GIFT   OF 


Mill- 


TEACHER'S  MANUAL 

*  AND  =======^^ 

COURSE  OF  STUDY 


[Based  upon  and  to  accompany  the  use  of  the"! 
California  State  Series  Texts  in  ArithmeticJ 


BY 


ALVA    WALKER    STAMPER 


251757 


I  UNIVERSITY 

**$*< 


PREFACE. 

This  course  in  arithmetic,  which  is  in  use  in  the  Training 
School  of  the  State  Normal  School,  Chico,  California,  has  been 
prepared  as  a  manual  for  the  schools  of  Chico  and  of  Butte 
County,  in  accordance  with  an  understanding  with  the  two 
boards  of  education  concerned.  The  size  of  the  edition  has,  in 
consequence,  been  limited  to  local  needs.  It  is  the  plan  of  the 
author  to  have  the  material  of  the  manual  appear  as  a  chapter 
in  his  "A  Text-book  in  the  Teaching  of  Arithmetic,"  which  will 
appear  early  in  1912.  In  this  book  a  system  of  cross  references 
will  be  used  between  the  chapter  referred  to  and  the  chapters 
on  method.  It  is  also  probable  that  this  manual  will  appear 
later  as  a  regular  bulletin  of  the  Chico  Normal  School. 

The  course  contained  in  this  manual  is  built  around  the 
central  idea  that  children  should  learn  to  "figure."  The  appli- 
cation side  of  arithmetic  has  been  given  consideration  but  with- 
out going  to  the  extreme  in  trying  to  vocationalize  the  subject. 
Perhaps  not  the  least  constructive  side  of  the  course  is  the  de- 
tailed work  outlined  for  the  first  two  years.  Inexperienced 
teachers  are  often  at  a  loss  to  know  how  to  systematize  their 
work  in  arithmetic  before  the  regular  drills  in  the  processes 
are  begun.  The  present  course  arranges  and  adapts  the  ma- 
terial in  Chapter  I.  of  the  State  Primary  Text  for  class  use. 
A  special  effort  has  been  made  to  unify  the  entire  course  with 
the  State  Series  Texts.  Teachers  will  find  it  necessary  to  learn 
thoroughly  the  primary  text's  special  plans  of  teaching  the  four 
fundamental  operations,  especially  addition  and  multiplication. 
There  is  no  half-way  method  of  using  the  book's  plan  of  be- 
ginning column  addition.  This  plan  and  other  features  of  the 
State  Texts  are  emphasized  in  this  course. 

ALVA  WALKER  STAMPER. 
State  Normal  School, 
Chico,  California. 

September,  1911. 


INTRODUCTION 


It  is  the  aim  in  the  first  year  and  a  half  to  give  the  founda- 
tions for  the  later  computation  work  and  the  applied  work  in 
denominate  numbers,  measurements,  and  problems.  Hence, 
there  is  a  place  in  this  early  work  for  (1)  counting,  recognizing 
number  groups,  reading  and  writing  numbers;  and  (2)  compar- 
ing, measuring,  and  related  work,  that  is  sometimes  classed 
under  sense-training.  There  should  be  familiarity  with  most 
of  the  addition,  subtraction,  multiplication,  and  division  facts 
in  connection  with  the  numbers  1 — 12,  or  possibly  1 — 20,  this 
knowledge  to  be  gained  through  the  use  of  objects.  No  at- 
tempt should  be  made  here  to  have  pupils  memorize  the  addi- 
tion combinations  and  other  number  facts,  the  idea  being  mere- 
ly to  acquaint  them  with  simple  applications.  The  most  im- 
portant thing  for  the  teacher  to  remember  in  this  preliminary 
work  is  that  great  care  must  be  used  to  give  pupils  the  neces- 
sary preparation  for  column  addition,  not  through  the  learning 
of  the  addition  combinations,  but  by  having  the  pupils  per- 
fectly familiar  with  the  natural  number  series,  so  that  they 
may  know,  for  example,  that  34  is  in  the  thirties,  that  it  comes 
before  35  and  follows  33,  that  it  is  the  next  number  after  29 
that  ends  in  a  4,  and  the  like. 

The  addition  and  subtraction  combinations,  column  addi- 
tion, and  subtraction  are  begun  in  the  High  Second  and  con- 
cluded in  the  High  Third.  The  multiplication  and  division 
tables,  mulplication,  and  division  are  begun  in  the  High  Third 
and  finished  in  the  High  Fourth.  The  special  plan  of  the  pri- 
mary text  in  teaching  the  combinations  and  tables  and  in  ap- 
plying them  requires  a  deferring  of  certain  combinations  and 
tables  until  they  are  needed  in  examples.  The  primary  text 
should  be  in  the  hands  of  the  pupils  in  the  Low  Third.  This  is 
advisable  since  many  of  the  exercises  are  especially  prepared 
for  class  use  by  the  pupils. 

Object  work  with  fractions  and  the  elements  of  denominate 
numbers  enter  into  the  earliest  work  under  the  aspect  of  com- 
parative magnitudes,  measuring,  and  the  like,  and  continue 


2  " '  '  '  • '  ARITHMETIC 

throughout  the  grades.  Many  problems  in  fractions  in  the 
form  of  partition  enter  into  the  problem  work  of  the  Fourth 
Grade.  Serious  work  in  common  fractions  is  undertaken  in 
the  Fifth  Grade,  all  the  processes  being  completed  with  the 
most  familiar  fractions.  Decimals  are  taken  up  and  finished 
in  the  High  Fifth,  orders  beyond  thousandths  being  deferred 
to  the  next  grade. 

The  advanced  text  is  introduced  in  the  Low  Sixth.  The 
work  of  the  Sixth  Grade  is  essentially  a  review  of  all  previous 
work,  the  aim  being  especially  to  give  the  pupils  added  facility 
in  the  use  of  common  and  decimal  fractions. 

The  Seventh  Grade  is  primarily  concerned  with  percentage 
and  its  applications  and  with  mensuration.  In  the  Low*ljighth, 
percentage  is  reviewed,  powers  and  roots  are  introduced,  and 
mensuration  is  completed.  This  concludes  the  standard  topics 
of  arithmetic.  The  work  of  the  High  Eighth  gives  a  review  of 
the  most  essential  parts  of  arithmetic  and  the  common  com- 
mercial aspects  of  the  subject,  including  the  keeping  of  simple 
accounts. 

The  one  thing  to  be  emphasized  most  throughout  the  grades 
is  addition.  If  the  plan  of  the  primary  text  is  followed,  the 
work  in  addition  should  be  in  excellent  shape  by  the  Low 
Fourth.  Teachers  should  see  to  it  that  the  facility  gained  by 
this  time  is  preserved  and  improved  upon.  It  is  only  by  con- 
tinued reviews  that  expertness  is  attained  and  maintained  in 
any  of  the  operations  of  arithmetic.  Teachers  should  provide 
for  reviews  throughout  every  term.  Ample  drill  should  be 
given  in  the  reading  and  writing  of  numbers,  especially  before 
taking  up  topics  requiring  this  facility.  Mental  arithmetic, 
which  is  often  neglected,  should  have  an  important  place  from 
the  time  the  first  number  facts  are  memorized  to  the  comple- 
tion of  the  course.  A  five-minutes  rapid  drill  in  mental  arith- 
metic at  the  beginning  of  the  period  in  the  upper  grades  will 
be  found  most  profitable.  Require  pupils  to  record  their 
answers  for  correction. 

The  early  use  of  language  forms  in  the  mechanical  work  is 
desirable.  This  practice  clears  up  ideas  for  the  pupils.  After 
the  teacher  is  sure  that  the  principles  in  any  operation  are  un- 
derstood, the  explanation  of  the  process  should  be  discontinued. 
The  first  mechanical  process  that  needs  the  use  of  a  language 
form  is  column  addition,  where  "carrying"  is  involved.  The 
pupil  is  taught  to  say,  for  example,  "Write  down  the  3  and  add 
the  1  to  the  next  lowest  figure." 

The  analysis  of  problems,  which  is  begun  in  the  Low  Fourth, 
is  made  easier  by  the  habits  of  correct  descriptions  of  processes 


ARITHMETIC  3 

gained  through  the  use  of  language  forms  in  the  earlier  grades. 
The  chief  purpose  of  the  explanation  is  to  make  plinciples  un- 
derstood. It  should  never  become  a  thing  in  itself.  Pupils 
should  work  the  great  majority  of  examples  and  problems 
without  explanations  of  any  sort. 

It  is  expected  that  teachers  will  follow  this  course.  If,  how- 
ever, it  seems  better  to  sacrifice  anything  in  order  to  cover  the 
main  topics  of  the  grade,  sacrifice,  for  example,  speed  but  not 
accuracy;  the  solution  and  analysis  of  the  harder  types  of 
problems  but  not  the  fundamental  processes  in  mechanical 
work;  the  longer  written  exercises  but  not  mental  arithmetic; 
and  the  short  processes  but  not  the  indispensable  longer  meth- 
ods in  common  use. 

Teachers  will  find  it  necessary  to  adhere  very  closely  to  the 
sequence  of  subject-matter  in  the  primary  text  on  account  of 
the  unique  plans  of  developing  the  addition  and  subtraction 
combinations  and  the  multiplication  and  division  tables.  There 
will  be  found  much  need  of  supplementary  work  for  the  pri- 
mary text.  Bulletin  No.  11,  issued  by  the  San  Francisco  State 
Normal  School  and  edited  by  D.  R.  Jones,  one  of  the  authors 
of  the  State  Texts,  will  be  found  very  helpful.  There  will  also 
be  found  need  of  supplementary  work  in  the  advanced  text, 
beginning  with  the  topic  of  percentage. 

A  number  of  texts  furnish  good  material  for  supplementary 
work,  among  these  being  the  Wentworth-Smith  Series  and  the 
Walsh  Series.  Among  texts  that  make  particular  efforts  to 
give  problems  having  significance  to  daily  living,  teachers 
should  refer  to  the  Smith  Series  and  to  the  Young  and  Jackson 
Series.  For  supplemental  work  in  mental  arithmetic,  Bailey's 
American  Mental  Arithmetic  is  good.  The  Speer  Text 
(Book  I.,  for  Teachers)  will  be  found  helpful  in  the  first  two 
grades  in  connection  with  sense-training  and  related  work. 


LOW  FIRST. 


The  chief  aim  is  the  preparation  for  column  addition,  which 
to  be  taught  a  year  and  a  half  later,  not  by  teaching  now  the 
addition  combinations  either  with  or  without  objects,  but  by 
teaching  the  children  to  know  the  places  of  the  numbers  in  the 
number  series.  This  presupposes  counting  and  a  knowledge 
of  the  number  symbols.  Much  attention  should  be  paid  to  the 
making  of  good  figures  and  writing  them  in  straight  columns. 
The  teacher  will  find  the  following  outline  of  work  for  the  Low 
First  arranged  so  as  to  give  an  easy  and  logical  development 
of  the  facts  to  be  learned  with  respect  to  the  place  of  numbers 


4  ARITHMETIC 

in  the  series.  This  arrangement  appears  below  under  the  cap- 
tions: Counting,  Number  Groups,  Reading  and  Writing  Num- 
bers, and  the  Place  of  Numbers  in  the  Series.  It  is  along  this 
line  that  pupils  are  to  be  systematically  drilled  throughout 
the  term. 

The  other  work  that  enters  in  the  Low  First  is  that  usually 
given  under  the  headings  of  sense-training,  busy  work,  meas- 
uring, and  comparisons.  The  order  in  which  this  is  introduced 
may  be  varied  by  the  teacher  as  occasions  demand.  It  is  in 
such  work  as  this  that  pupils  learn  to  express  their  thoughts 
in  simple  language  based  upon  experiences  gained  within  or 
without  the  class.  Lessons  I. -XIII.  in  Chapter  I.  of  the  primary 
text  furnish  material  for  much  of  this  work.  These  les- 
sons from  the  text  should  not  be  taught  in  the  order  there 
given.  Neither  should  all  of  some  of  the  lessons  be  taught  in 
this  grade.  The  assignments  below  explain  this.  Only  the 
simplest  phases  of  the  matter  treated  should  be  introduced  in 
the  Low  First.  Teachers  should  not  be  misled  into  magnifying 
the  extent  of  the  work  expected  in  this  line  on  account  of  the 
variety  of  topics  presented  below.  Of  the  two  lines  of  work 
mentioned  in  this  and  the  preceding  paragraph,  the  first  should 
receive  chief  attention.  Much  time  is  needed  especially  in 
writing  numbers. 

1.  Relative  Position,  Direction,  Magnitude:     Call  on  individual 
pupils  and,  in  certain  exercises,  the  whole  class.     Which  is 
your  right  hand?    Which  is  your  left  hand?  Your  right  eye? 
Place  your  left  hand  on  your  right  arm.    Who  is  on  the  right 
of  Mary?     On  the  left  of  John?     Point  to  your  right.     Your 
left.     Behind  you.     In  front  of  you.     Write  the  figure  2  on 
the  board.    Write  another  2  above  the  first  2.    Write  another 
below  the  first  2.    Write  a  1  on  the  board.    Write  another  1 
close  to  this  and  to  the  right  of  it.     Write  another  1  close 
to  the  first  1  and  to  the  left  of  it.     Point  to  the  top  of  the 
blackboard.     Point  to  the  bottom  of  the  blackboard.     Mary 
will  stand  to  the  left  of  John.    Julia  will  stand  between  John 
and  Mary.     Place  your  finger  on  the   middle  of  this  ruler. 

Which  of  these  blocks  is  the  largest?  Which  is  the 
smallest?  Which  is  longer,  this  ruler  or  that  ruler?  Who  is 
the  tallest  pupil  in  the  class?  Who  is  the  shortest?.  Who 
is  taller,  Mary  or  John?  Who  is  shorter,  Ida  or  Sue?  The 
class  will  stand  in  a  row.  Who  is  the  tallest?  He  may 
stand  first  in  the  row.  Who  is  next  to  the  tallest  in  the 
class?  He  may  next  take  position.  The  pupils  take  position 
according  to  their  respective  heights.  Compare  weights  of 
objects  at  hand.  See  Lesson  I.  in  text  and  Speer,  Book  I. 
for  Teachers. 

2.  Counting:      1  to  12;     1   to  20.     With    and    without    objects. 
There  is  little  need  of  counting  with  objects  beyond  20.  Have 
pupils  co,unt  things  they  are  interested  in.  How  many  pupils 


ARITHMETIC  5 

in  the  class  this  morning?  How  many  boys?  How  many 
girls?  How  many  feet  has  a  horse?  Name  other  animals 
that  have  four  feet.  John  may  make  three  marks  on  the 
board.  Mary  may  make  some  marks  on  the  board.  Sue 
may  tell  how  many  marks  Mary  made.  See  that  the  place 
of  number  in  the  series  is  not  confused  with  number  itself. 
Thus,  ask  Mary  to  fetch  the  first  three  of  a  number  of 
blocks  placed  in  a  row.  Then  ask  her  to  replace  them  and 
bring  the  third  block  in  the  row.  Who  is  the  third  pupil 
from  the  left  in  line?  Who  is  the  second  from  the  other 
end?  The  abacus  furnishes  a  convenient  means  for  count- 
ing with  objects.  See  first  part  of  Lesson  II.  of  the  text. 

3.  Number  Groups:     Unless  specially  arranged,  pupils  cannot, 
recognize  more  than  four  or  five  objects  in  a  group.     Teach 
the  recognition  of  number  groups  as   on  dominoes.     Have 
flash  cards  with  spots  thus  arranged.     Place  the  pupils  two 
in  a  group  and  have  them  count  the  number  of  groups.     The 
abacus  is  handy  for  this  kind  of  work,  where  the  aim  is  to 
prevent  the  idea  of  a  fixed  unit.    How  many  3's  do  you  see? 
How  many  2's?     How  many  1's?     This  refers  to  objects  in 
groups,  not  to  the  number  symbols.    Do  not  yet  count  serial- 
ly by  2's,  3's,  etc. 

4.  Reading    and    Writing    Numbers:      1   to   12;    1   to  20.     Board 
work  only.     No  pencil  throughout  the  term.     A  large  class, 
only,  should  excuse  its  use.     Relate  the  number  (found  by 
counting),   the   name,   and   the   symbol    (the   figure).     Read 
numbers  written  by  the  teacher.     Use  flash  cards  to  teach 
quick  recognition  of  the  figures. 

Pupils  write  numbers  at  board  under  models  set  by  the 
teacher.  The  crayon  should  be  held  not  tightly,  yet  firmly. 
Short  pieces  of  crayon  are  desirable  for  small  hands.  Aim 
toward  large  rather  than  small  figures.  Where  pupils  have 
difficulty  let  them  trace  over  the  copy  made  by  the  teacher. 
Then  try  without  tracing.  The  teacher  makes  a  3  on  the 
board.  She  asks  the  class  to  look  at  it  carefully.  She  erases 
the  3  and  asks  the  class  to  make  one  like  it.  Write  13  on 
the  board.  Which  figure  is  on  the  right  side?  Which  is  on 
the  left  side?  Which  figure  do  we  make  first  in  writing  13? 
In  reading  the  "teens"  the  teacher  should  place  the  pointer 
first  on  the  right  figure  as  the  word  is  pronounced.  This 
will  help  prevent  the  writing  of  13  with  the  3  first.  Much 
individual  attention  should  be  given  the  pupils  while  they 
are  learning  to  make  the  different  figures.  Be  sure  that  the 
8  is  made  toward  the  left,  beginning  at  the  top.  Write  the 
numbers  in  columns,  vertical  and  straight. 

5.  Counting:      1  to  50. 

6.  The  Place  of  the  Numbers  in  the  Series:     1  to  10;   1  to  20. 
Extend  the  limits  of  the  series  as  indicated.     The  teacher 
should  have  on  the  board  the  series  to  be  studied  written 
from  the  bottom  up  (to  prepare  for  the  special  plan  of  the 
primary  text   in  teaching  column  addition).     The  numbers 
1  to  50  may  be  printed  on  strong  linen  or  heavy  paper.     A 
strip  9  feet  long  and  6  inches  wide  will  answer  the  require- 
ments for  figures   1  5/8   inches  high.     A  set  of  stamps  for 
figures  of  this  size  costs  about  50  cents.     The  chart  is  to  be 


6  ARITHMETIC 

used  in  the  following  exercises.  Do  not  force  the  children 
to  visualize  the  figures.  With  the  chart  on  the  wall  for 
their  daily  study,  the  power  of  visualizing  will  follow  easily 
and  naturally. 

a.  Name  the  number  next  after  3,  7,  etc.;  and,  later,  after 
13,  19,  etc. 

b.  What  number  before  6,  10,  etc.?;   and,  later,  before  13, 
16,  20,  etc.? 

c.  Name  the  number  between  5  and  7,  12  and  14,  etc. 

d.  Name  the  numbers  between  4  and  7,  15  and  18,  etc. 
7.     Reading  and  Writing  Numbers:     1  to  50. 

8  Time  Measures:  What  day  is  today?  What  day  was  yes- 
terday? What  day  will  tomorrow  be?  Name  the  days  of 
the  week.  How  many  days  in  one  week?  How  many  school 
days?  How  many  working  days?  Y/hat  month  is  this?  In 
what  month  is  your  birthday,  Sara?  What  day  of  the  month 
(date)  is  your  birthday?  What  day  of  the  month  is  today? 
At  what  hour  in  the  morning  does  school  begin?  When 
does  school  let  out  in  the  morning?  What  time  does  school 
open  in  the  afternoon?  Refer  to  the  clock  face.  The  teacher 
draws  a  figure  of  a  clock,  using  Arabic  numerals.  Where 
is  the  hour  hand  at  9  o'clock?  Where  is  the  minute  hand? 
Sara  was  5  minutes  late  this  morning.  Where  was  the  min- 
ute hand?  Do  not  go  beyond  the  "time"  that 'is  of  interest 
to  the  pupils.  Compare  the  ages  of  the  pupils. 

9.  Sense-Training:     Review  relative  size,  as  previously  taken 
up,  thus  training  the  sight  judgment  of  the  pupils.     Train 
judgment  of  relative  size  through  the  sense  of  touch,   the 
eyes  being  closed.     For  both  of  these  purposes  use  various 
kinds  and  shapes  of  objects.     Compare  weights  of  objects. 
Thus  far  the  pupils  are  not  supposed  to  make  exact  com- 
parisons, although  the  "one-half"  and  "two  times"  may  be 
brought  out.     Teach  Forms  as  in  Lesson  IX.  of  the  text. 

Train  the  pupils  in  the  power  of  visualizing  objects,  the 
aim  being  to  have  them  recall  the  properties  of  the  objects 
and  the  number  facts  involved.  Write  a  number  on  the 
board.  Erase  it  and  ask  for  the  number  erased.  Write  two 
numbers  on  the  board.  Erase  them  and  ask  for  the  num- 
bers erased.  Ask  questions  about  the  position  of  the  figures 
in  certain  numbers  and  see  if  the  pupils  have  the  correct 
mental  picture.  Test  with  13  and  31.  See  Speer,  pp.  37-48, 
for  work  in  sense-training. 

10.  Counting  and  Reading  and  Writing  Numbers:     1  to  50.    Re- 
view previous  work.    How  many  figures  in  25?    In  6?    Have 
pupils  point  out  numbers  on  the  chart  as  they  count.    Point 
out  on  the  chart  numbers  that  have  been  named.    In  looking 
for  43,  for  what  number  do  you  look  first?     Point  out  num- 
bers on  the  chart  and  have  them  named.     How  many  num- 
bers  written   on   the   chart?     Children   count   and   discover 
that  the  last  number  written  in  the  series  tells  how  many 
have  been  written. 

Ask  the  pupils  to  write  the  first  5,  10,  or  15  numbers  in  a 
column,  this  to  be  done  without  reference  to  the  chart.  Ask 
them  to  begin  with  a  certain  number  and  write  two,  three, 
five,  or  ten  more.  Count  by  10's  and  5's. 


ARITHMETIC  7 

11.  Comparative  Magnitudes:  Review  previous  work.  Com- 
pare both  unequal  and  equal  magnitudes,  using  a  great  va- 
riety of  objects  and  drawings.  Most  of  the  drawing  should 
be  done  by  the  teacher  and  on  the  board,  the  square,  rec- 
tangle, and  circle  being  employed.  The  pupils  may  be  given 
a  number  of  blocks  of  equal  sizes,  from  which  they  may 
build  larger  solids  that  will  furnish  means  of  comparison. 
Colored  squares  made  out  of  cardboard  will  interest  the 
children  and  serve  a  similar  purpose.  These  objects  and 
drawings  will  give  sufficient  material  for  bringing  out  the 
relation  "one-half"  and  "two  times,"  and  perhaps  a  few 
other  simple  fraction  facts. 

Develop  the  need  of  measuring  to  determine  equality.  Re- 
alte  here  the  foot  and  the  yard  (Numbers  1  to  3)  and  the 
inch  and  the  foot  (Numbers  1  to  12).  See  Lessons  IX.,  VI. 
(first  part),  and  VII.,  in  the  text.  Refer  to  Speer,  but  keep 
in  mind  that  the  work  in  comparing  magnitudes  there  given 
extends  over  more  ground  than  is  contemplated  for  the  Low 
First.  Most  of  the  material  on  pp.  37  to  60  of  that  text  can  be 
used  here. 

12.. The  Place  of  the  Numbers  in  the  Series:  1  to  50.  Use  the 
chart  with  numbers  written  from  the  bottom  up.  Encourage 
the  pupils  in  visualizing  the  numbers  on  the  chart.  After 
the  position  of  the  numbers  is  fixed  in  his  mind,  the  pupil 
naturally  can  answer  the  teacher's  questions  more  readily 
when  not  looking  at  the  chart  than  if  required  to  examine  it. 

a.  Drill  as  in   (a),   (b),   (c)   under   (6)   above.     Emphasize 
especially  asking  for  the  numbers  next  after  9,  19,  29,  39,  49, 
and  the  numbers  immediately  before  50,  40,  30,  20,  10. 

b.  The    decades.      After    the    first    ten   numbers   are   the 
"teens."      After    the    "teens"    are    the    twenties.      After    the 
twenties  are  the  thirties,  etc.     Before  the  twenties  are  the 
"teens."    Before  the  thirties  are  the  twenties.    Etc. 

c.  Step  half-way  across  the  room  and  then  stop,  Charlie. 
Locate  the  middle  of  this  stick,  Mary.     Point  out  the  half- 
way point  of  this  vertical  line  on  the  board,  Arthur.     Find 
the  number  on  the  chart  that  is  half-way  between  10  and  20. 
That  is  half-way  between  20  and  30.     Etc.     Is  23  nearer  20 
or  30?     Is  37  nearer  30  or  40?     Etc. 

The  number  table  may  be  extended  to  100  for  counting, 
reading  and  writing  numbers,  and  locating  numbers  in  the 
series  in  case  the  pupils  are  ready.  The  counting  and  read- 
ing and  writing  numbers  to  100  can  be  done  almost  as  soon 
as  the  pupils  can  do  this  to  50,  but  the  locating  the  num- 
bers in  the  series  50  to  100  may  better  be  left  for  the  High 
First. 


HIGH    FIRST. 


The  work  of  the  High  First  is  a  review  and  continuation  of 
the  work  of  the  previous  term.  The  main  emphasis  should  be 
on  the  reading  and  the  writing  of  numbers  and  locating  num- 
bers in  the  series.  The  study  of  comparative  magnitudes  is 
continued.  The  essentially  new  work  consists  of  finding  out 


8  ARITHMETIC 

the  number  facts  in  connection  with  objects.  In  this  connec- 
tion the  numbers  1 — 12  are  studied,  the  pupils  employing  addi- 
tion, subtraction,  multiplication,  and  division,  the  latter  in  its 
dual  form,  division  by  measuring  and  division  by  partitioning. 
No  effort  is  to  be  made  here  to  teach  the  abstract  number  facts, 
for,  according  to  the  plan  of  the  course,  no  use  is  to  be  made 
of  these  facts  for  a  year  to  come.  But  it  is  essential  that  pupils 
learn  the  meaning  of  these  processes  in  connection  with  things 
and  hence  the  value  of  the  work  planned  for  this  grade.  Fur- 
thermore, we  may  remind  ourselves  here  that  an  objective  un- 
derstanding of  9  and  4  are  13  bears  no  immediate  relation  to 
the  memorizing  of  that  fact.  When  the  pupil  is  ready  to  use  the 
fact  that  9  and  4  are  13  he  should  not  be  bothered  then  with 
the  so-called  reason  involved.  He  needs  to  use  it  then  as  a 
tool.  Number  stories  should  accompany  this  early  study  of 
the  operations. 

The  abstract  drills  indicated  below  in  connection  with  rec- 
ognizing the  place  of  the  numbers  in  the  series  should  be  given 
in  the  order  as  arranged.  The  rest  of  the  work  may  readily  be 
rearranged  according  to  the  wish  of  the  teacher.  Before  teach- 
ing any  topic,  review  the  corresponding  work  in  the  Low  First. 

1.  Counting:      1  to  100;    1  to  120.     Count  by  1's,  10's,  5's,  and, 
later,  by  2's.     Introduce  even  and  odd  numbers.     Count  by 
naming  the  even  numbers  from  2  to  20;   20  to  50.     Similarly 
with  the  odd  numbers.     Count  objects,  naming  the  place  in 
the  series.     Thus,   1st,  2nd,  3rd,  etc. 

2.  Reading  and  Writing   Numbers:      1  to  100;    1  to  120.     Much 
time  should  be  spent  on  this  work.    It  is  preferable  to  write 
numbers   entirely   on   the  board.     The   class   may  write   37. 
Yours  is  good,  John.     You  may  now  make  a  straight  column 
of  three  37's  and  be  careful  not  to  write  too  near  the  vertical 
line.     Correct  individual  errors  of  the  pupils.  Write  the  even 
numbers  up  to  20.     The  odd  numbers  up  to  21.     Yours  is  not 
good,  Julia,  because  your  second  column  is  too  near  the  first. 
Write  the  numbers  1  to  20,  placing  the  odd  numbers  on  the 
left  of  a  vertical  line  and  the  even  numbers  on  the  right. 

3.  Time  Measures:      Extend  the  work  of  the  Low  First.     Tell 
time  to  the  nearest  5  minutes,  first  reviewing  counting  by 
10's  and  5's  as  far  as  60.     Teach  the  reading  (not  writing) 
of  the  Roman  numerals,  I.  to  XII.     Count  5  seconds,  10  sec- 
onds.    Remain  silent  10  seconds.     Point  out  on  the  clock  5 
minutes    after    the    hour.      10    minutes.      Etc.      How    many 
minutes  in  one  hour?     In  a  half  hour?     In  a  quarter  of  an 
hour?     How  long  does   it  take  you   to   walk  home?     How 
much  time  do  we  take  for  recess?    For  noon? 

4.  The   Place  of  the    Numbers   in   the   Series:      1   to   100;    and, 
later,  1  to  120.     Use  the  charts  referred  to  in  the  work  of 
the  Low  First  and  follow  the  plan  of  (6)  and  (12). 

5.  Comparative   Magnitudes:      Bring  out  objectively  the   ideas 


ARITHMETIC  9 

of  1/2,  1/4,  and  1/3.  Draw  diagrams  of  squares  and  rec- 
tangles and  compare  lengths  of  sides.  Compare  areas 
(sizes).  Also  use  blocks.  Speer  has  good  suggestions.  Give 
the  children  squares  of  paper.  Give  directions  for  folding 
so  as  to  get  two  equal  parts;  to  get  four  equal  squares. 
Bring  out  the  1/2  and  1/4  relations.  Also  twice  as  large  and 
four  times  as  large.  Use  Lesson  X.  of  the  text,  omitting  for 
the  present  the  use  of  fractions  in  connection  with  num- 
bered objects,  such  as  in  the  phrase,  1/2  or  10  balls. 

6.  Counting:  1  to  120.  Emphasize  the  decades.  Count  by  10's, 
beginning  with  10,  1,  2,  3,  etc.  For  this  purpose  have  a  num- 
ber table  already  written  on  the  board,  the  numbers  1  to  10 
in  the  first  column,  11  to  20  in  the  second,  21  to  30  in  the 
third,  and  so  on  up  to  100.  Read  the  horizontal  rows.  Then 
count  without  looking  at  the  table.  See  Lesson  II.  and  the 
first  part  of  Lesson  X.  in  the  text. 

7  The  Place  of  the  Numbers  in  the  Series:  1  to  120.  Use  the 
charts  as  an  aid  toward  visualization. 

a.  First  explain,  for  example,  what  is  meant  when  we  say 
that  24  ends  in  a  4  or  that  16  ends  in  a  6.    What  does  34  end 
in?    47?    50?    Find  a  number  among  those  that  I  write  that 
ends  in  a  3?     In  a  9? 

b.  What  number  in  the  twenties  ends  in  a  3?    In  the  thir- 
ties that  ends  in  an  8?    In  the  "teens"  that  ends  in  a  4?  Etc. 

c.  What  is  the  first  number  after  20  that  ends  in  a  0?  Etc. 
What  is  the  first  number  after  5   that  ends   in  a  5?     Etc. 
Similarly  with  other  endings  ten  numbers  apart. 

d.  Bring  out  the  ideas  of  greater  than  and  less  than  (after 
and  before  the  series).    Which  comes  first,  15  or  17,  26  or  29, 
38  or  40,  31  or  29?     Notice  the  gradation.     The  last  set  of 
numbers  carries  the  pupil  into  different  decades.     Which  is 
greater,  12  or  10,  17  or  14,  23  or  27?     Etc. 

e.  Numbers  after  10,   20,   etc.,  that  end   with  certain   fig- 
ures but  not  ten  numbers  apart.     What  is  the  next  number 
after  10  that  ends  in  a  3,  5,  etc.?    The  same  after  20,  30,  etc.? 

f.  Numbers  after  any  numbers  in  the  same  decade.    What 
is  the  next  number  after  13  that  ends  in  a  5,  7,  etc.?    After 
23,  etc.? 

g.  Numbers  after  other  numbers,  the  latter  ending  in  0. 
What  is  the  next  number  after  16  that  ends  in  a  0?     After 
26?    Etc. 

h.  From  one  decade  into  the  next.  What  is  the  next  num- 
ber above  7  that  ends  in  a  1?  Above  17  that  ends  in  a  1? 
Above  27,  37,  etc.?  The  same  with  other  endings.  This 
drill  is  very  important  since  the  ideas  involved  are  essential 
in  later  work  in  column  addition. 

8.  Complementary  and  Measure  Contents  of  Numbers:  Num- 
bers 1  to  6;  1  to  12.  This  work,  in  which  the  pupils  are  to 
learn  the  early  number  facts,  is  to  be  entirely  objective.  Use 
all  sorts  of  objects.  Splints  blocks,  and  the  balls  of  abacus 
are  good.  It  is  perhaps  better  to  teach  the  complementary 
contents  for  any  number,  that  is,  the  addition  and  subtrac- 
tion facts,  first;  but  in  concrete  work  with  objects  where  no 
effort  is  made  to  have  the  pupils  memorize  results,  teachers 
will  find,  for  example,  that  a  pupil  can  find  the  number  of 


10  ARITHMETIC 

2's  in  6  as  readily  as  he  can  show  the  teacher  with  his  sticks 
that  2  and  4  are  6.  The  addition  and  subtraction  facts  are 
to  be  taught  for  all  numbers  in  the  limits  assigned  above. 
The  measure-contents'  facts,  that  is,  multiplication,  division, 
and  partition,  are  to  be  taught  in  close  connection  with  the 
complementary-contents'  facts,  but  do  not  at  this  time  study 
the  prime  numbers,  3,  5,  7,  and  11. 

As  a  plan  of  procedure,  the  following  is  suggested:  Teach 
the  complementary  contents  of  the  number  6.  Here  the  pupil 
finds  from  the  use  of  objects  that  5  and  1,  1  and  5,  4  and  2, 
2  and  4,  and  3  and  3  make  6;  and  that  subtracting  in  turn 
1,  2,  3,  4,  5,  from  6  leaves  5,  4,  3,  2,  and  1,  respectively.  Then 
teach  the  measure  contents  of  6.  Thus  two  3's  are  6,  three 
2's  are  6;  in  6  there  are  two  3's  and  three  2's;  1/2  of  6  is  3, 
and  1/3  of  6  is  2.  There  is  no  gain  teaching  now  that  six 
1's  are  6,  one  6  is  6,  and  1/6  of  6  is  1.  Next  teach  the  meas- 
ure contents  of  4.  Next  the  complementary  contents  of  the 
number  7.  Then  all  the  number  facts  about  8  in  the  way 
number  6  was  studied.  Take  up  the  other  numbers  as  the 
term  advances  in  the  way  that  seems  best  to  the  teacher. 

Teach  number  stories  in  connection  with  the  above  work. 
This  gives  the  pupils  a  chance  to  cultivate  their  powers  of 
expression.  The  number  story  should  follow  the  objective 
presentation  and  be  based  upon  it. 

Teach  Lessons  III.,  IV.,  and  VI.  of  the  text,  which  are  les- 
sons in  comparative  magnitudes,  in  connection  with  the 
above  work.  Lesson  X.  gives  exercises  in  partition,  pre- 
viously omitted.  Use  Lesson  XII.  for  suggestions  in  giving 
number  stories. 

9.  Comparative   Magnitudes:      Teach  Lesson  VIII.  of  the  text, 
on  pints,   quarts,   and   gallons,   if  not   already  used   in    (8). 
Have  the  necessary   measures   in   class   and  let   the   pupils 
compare  capacities  by  filling  with  water.     Review  measure- 
ments, using  feet,  inches,  and  yards.     Let  the  children  make 
drawings,  dividing  up  squares,  rectangles,  and  the  like. 

10.  Counting,   Reading,  and   Writing    Numbers:      1  to  200;    1  to 
1000.     Omit  teaching  of  place  value  with  the  exception  of 
naming  and  having  the  pupils  name  hundreds'  place.     Count 
by  10's  and  100's.     Write  by  10's  and  100's. 


LOW  SECOND. 


Review  the  work  of  the  previous  grades.  By  the  end  of 
this  grade  the  pupils  should  acquire  considerable  command  of 
oral  expression  in  the  giving  of  number  stories  and  the  like. 
The  placing  of  the  numbers  in  the  series  should  be  continued 
and  emphasized  even  more  than  in  the  previous  grade,  since 
in  the  next  grade  column  addition  will  be  begun  . 

1.  Counting,  Reading,  and  Writing  Numbers:  1  to  10..000.  Con- 
tinue serial  counting  as  in  the  previous  grade.  Count  by 
10's,  5's,  2's  (both  the  series  of  the  odd  and  the  even  num- 
bers), 100's,  and  1000's.  Begin  at  any  multiple  of  10  in  the 
series  to -count  in  this  manner.  Thus,  begin  with  360  and 


ARITHMETIC  11 

count  by  10's  to  400.  Count  by  100's  from  300  to  1000.  Count 
by  1000's  from  1000  to  10,000.  Write  the  higher  numbers  in 
the  natural  serial  order  and  as  above.  Spend  much  time  on 
the  reading  and  writing  of  the  higher  numbers.  Have  both 
board  and  seat  work.  Teach  the  names  of  the  different  or- 
ders, or  places.  By  the  end  of  the  term  the  pupils  should 
learn  how  to  build  numbers  of  two  places  and  of  three 
places  by  using  splints  and  rubber  bands.  Teach  all  of  Les- 
sons V.  and  II.  of  the  text. 

2.  Comparative    Magnitudes:      Extend    the    drawing   work   for 
fractional    and    multiple    relations    begun    in    the    previous 
grade.     Let  the  pupils  draw  diagrams  freehand  at  the  board 
and  with  short  rulers  at  the  seat  showing  relations  specified 
by  the  teacher.     Thus,  draw  a  square.     Divide  it  so  as  to 
show  1/4  of  it.     Again,  draw  a  square  and  then  a  second 
square  near  the  first  that  shall  be  4  times  the  size  of  the 
first  square.     The  fraction  symbols  may  now  be  introduced 
but  are  not  used  in  operations.    Teach  the  1/2,  1/4,  1/3.    The 
pupils  learn  the  relations  that  exist  between  1/4,  2/4,   1/2, 
3/4,  and  4/4.     Also  between  1/3,  2/3,  and  3/3;   and  between 
2/2,  3/3,  and  4/4.     Relate  to  partition  (See  (4)  below)  with 
counted   objects.     Thus,   find   1/3   of   12   counters;    2/3   of   6 
blocks. 

3.  Place  of  the   Numbers  in  the  Series:     Review  the  work  of 
the  previous  grade,  extending  the  numbers  beyond  120.  The 
number  charts  (1  to  50;  50  to  100)  should  be  on  the  wall  for 
reference.     Among  other  things  to  emphasize  is  the  naming 
of  the   number  following     another     number  and  having  a 
stated  ending.     Thus,   give  the  next  number  after  83  that 
ends  with  a  1.     This  is  vital  in  column  addition,  for,  when 
the  pupil  has  learned  that  3  and  8  are  11  and  that  8  added 
to  any  number  ending  in  a  3  gives  a  number  ending  in  a  1, 
he  is  in  a  position  to  tell  that  91  is  the  sum  of  83  and  8. 

4.  Complementary  and  Measure  Contents:     Review  and  extend 
the  work   of  the   previous   grade.     Carry  the   study   of  the 
numbers,  still  entirely  objectively,  to  20.     Make  no  attempt 
to  have  facts  memorized.    Work  of  this  kind  is  not  intended 
so  much  to  prepare  for  the  later  learning  of  the  number 
facts,  because  this  would  be  idle,  but  it  is  given  so  as  to  af- 
ford the  pupils  a  first-hand  application  of  the  four  funda- 
mental operations  of  arithmetic  to  things.     See  the  sugges- 
tion at  the  end  of  (2)  above.     In  teaching  the  measure  con- 
tents, consider  first  exact  multiples  and  divisors.     Thus,  in 
12  there  are  three  4's.     Later,  in  14  there  are  three  4's  and 
2  "over."     Continue  the  use  of  number  stories.     While  it  is 
suggested  that  the  work  under  this  topic  be  carried  as  far 
as  20,  it  is  not  intended  that  the  work  should  be  made  scien- 
tifically exhaustive.     The  teacher  should  keep  in  mind  above 
all  the  interest  of  the  children.     The  abacus  will  be  found 
very  helpful  in  such  work.     Teach  Lessons  XI.  and  XIII.  in 
the  text.  * 

5.  Addition  and   Subtraction   That   Is   Permissible  with   Count- 
ing:    It  is  generally  agreed  that  children  should  not  add  by 
counting.     Thus,  in  the  next  grade  they  will  learn  that  4 
and  5  are  9  merely  a.s  a  fact,  not  being  permitted  to  count 


12  ARITHMETIC 

5  more  beyond  4  to  get  the  result.  The  addition  and  sub- 
traction of  1  and  10  and  5  as  shown  below  is  permissible  and 
desirable: 

a.  Add  and  subtract  1.     What  are  9  and  1,  19  and  1,  29 
and  1,  etc.?    What  is  1  less  than  10,  20,  30,  etc.?    What  are 
4  and  1,  14  and  1,  24  and  1,  etc.?    What  is  1  less  than  5,  15, 
25,  etc.?     Then  give  miscellaneous  eexrcises  without  refer- 
ence to  the  decades — 7  and  1,  23  and  1,  87  and  1;  1  less  than 
7,  16,  93. 

b.  Adding  to  numbers  ending  with  0  with  the  correspond- 
ing subtractions.     What  are  10  and  3,  20  and   3,  30  and  3, 
etc.?     What  are  10  and  9,  20  and  9,  etc.?    What  are  10  and 
10,  20  and  10,  30  and  10,  etc.?     What  is  13  less  3,  23  less  3, 
33  less  3,  etc.? 

c.  Adding  10  to  numbers  with  the  corresponding  subtrac- 
tions.   What  are  3  and  10,  13  and  10,  23  and  10,  etc.?    What 
is  10  less  than  13,  23,  33,  etc.? 

d.  Adding  5  to  numbers  ending  with  a  5  with  the  corre- 
sponding subtractions.     What  are  5  and  5,  15  and  5,  25  and 
5,  etc.?    What  is  5  less  than  10,  20,  30,  etc.? 

All  the  above  questions  are  based  on  the  pupil's  previous 
experience  in  serial  counting. 

Also  associate  1  and  9  with  9  and  1,  1  and  4  with  4  and  1, 
etc.  Emphasize  the  making  up  idea  in  subtraction,  which 
is  used  later  when  subtraction  is  systematically  taught. 
Thus  ask,  9  and  what  are  10?  4  and  what  are  5?  20  and 
what  are  25?  Etc. 

In  the  review  work  of  this  term  that  relates  to  the  text, 
review  especially  Lessons  VII.,  VIIL,  X.,  and  XI.  The  pupils 
should  have  covered  the  material  of  Lessons  I. -XIII. 


HIGH  SECOND. 

Text. — California  State  Series,  "First  Book  in  Arithmetic,"  pp. 
34-56.  Supplementary  exercises  from  Bulletin  No.  11,  Hand- 
book for  Teachers,  Part  I.,  State  Normal  School,  San  Fran- 
cisco, Cal.,  edited  by  D.  R.  Jones.  Text  in  the  hands  of  the 
teacher. 

Beginning  Work  in  Formal  Processes. — Addition  and  subtrac- 
tion combinations  of  Lesson  A  of  the  text.  Examples  in 
addition  and  subtraction  to  correspond  to  the  combinations 
learned.  Defer  the  hard  cases  in  subtraction. 

Language  Forms  and  Technical  Terms. — Oral  language  forms 
in  concrete  problems  and  in  column  addition  where  "carry- 
ing" is  involved.  Use  of  terms  sum  and  difference. 


Approximately  the  first  half  of  the  High  Second  should  bv 
given  to  a  complete  review  of  the  previous  one  and  one -half 
years.  If  the  previous  work  in  counting  ,in  reading  numbers, 
and  in  locating  numbers  in  the  series  has  been  well  done,  this 
part  of  the  review  should  be  easily  and  quickly  done.  Continue 
to  perfect  the  making  of  good  figures  and  writing  them  in 
straight  columns.  It  is  sufficient  to  have  the  pupils  read  and 


ARITHMETIC  13 

write  numbers  up  to  10,000,  the  limit  assigned  for  the  Low  Sec- 
ond.   Pages  43-46  of  Chapter  III.  of  the  text  fit  in  well  with  the 
review.     The  review  in  the  study  of  comparative  magnitudes 
and   the    complementary   and    measure    contents    of   the   early 
numbers,  still  in  connection  with  objects,  should  receive  con- 
siderable  attention.     The   responses   of   the   children   in   story 
problems  should  now  be  readily  given.     This  review  should  in- 
clude the  material  of  Chapter  I.  of  the  text. 
Place  Value:     The  pupils  should  already  know  the  names  of 
the  orders  in  numbers  having  four  figures.     Test  them  by 
questioning.     Which  figure  in  3,579   is  in  hundreds'   place? 
Which  figure  is  in  tens'  place?     In  units'  place?    In  thous- 
ands' place?    In  5,642,  which  place  does  the  2  occupy?     The 
6?    Etc.     Enumerate  from  the  right,  saying  units,  tens,  etc. 
Have  much  practice  in  reading  numbers  and  in  answering 
questions  like  the  above.     The  proper  placing  of  the  figures 
when  writing  numbers   should   now  be   more  readily  done. 
As  a  preliminary  to  work  like  the  above  or  parallel  with 
it,  have  the  pupils  build  numbers  with  splints,  using  rubber 
bands  to  hold  the  bundles  of  tens  together.    Build  32.    Write 
the  number  on  the  board.     Identify  the  3  in  the  written  32 
with  the  3  bundles  of  tens.     In  this  exercise  the  children, 
not  knowing  in  advance  that  32  is  being  built,  write  the  fig- 
ures that  represent  the  number  of  splints  used.     Next  write 
a  number  on  the  board  and  have  it  built.     Build  numbers 
of  three  figures. 

United  States  Money:  See  that  the  pupils  can  recognize  the 
following  coins:  5c,  lOc,  25c,  50c,  and  $1.  What  other  name 
has  25c?  50c?  If  a  whole  cake  is  worth  $1,  what  is  1/2  of  it 
worth?  What  is  1/4  of  it  worth?  If  a  whole  cake  is  worth 
50c,  what  is  1/2  of  it  worth?  Identify  75c  as  3/4  of  a  dollar. 
What  may  be  purchased  with  any  of  the  above  amounts? 
Have  the  children  assemble  the  coins  to  show  15c,  20c,  30c, 
etc. 

Make  change  as  the  clerk  in  the  store  does.  This,  of 
course,  must  involve  only  those  number  facts  that  the  chil- 
dren have  memorized,  namely — the  adding  and  subtracting 
of  10  and  5.  For  this  purpose,  toy  money  may  be  used.  Ex- 
ercises like  the  following  are  within  the  powers  of  the 
pupils:  I  buy  some  candy  worth  5c.  I  give  the  clerk  lOc. 
He  gives  me  5c  in  change.  I  buy  a  pound  of  coffee  worth 
50c.  I  give  the  clerk  $1.  He  gives  me  50c  in  change.  I  buy 
lOc  worth  of  apples.  I  give  the  clerk  25c.  He  gives  me  15c 
in  change.  The  teacher  may  play  the  clerk  and,  as  in  the 
last  example,  will  give  back  the  change  as  follows:  She 
says  lOc.  Then  she  lays  out  lOc,  saying  20c.  She  then  lays 
out  5c  more,  saying  25c.  The  pupils  should  also  be  taught 
to  do  this. 

Preparation  for  Column  Addition:  The  drills  on  the  placing 
of  the  numbers  in  the  series,  begun  in  the  Low  First  and 
extending  into  the  High  Second,  have  been  planned  with 
special  reference  to  needs  in  column  addition.  Some  of  the 
most  important  things  emphasized  have  been: 
a.  Counting  by  1's,  10's,  and  5's.  Counting  by  10's,  begin- 


14  ARITHMETIC 

ning  with  any  number.    Counting  the  series  of  even  and  odd 
numbers. 

b.  Naming  the  number  after  and  before  a  number.    Nam- 
ing the  numbers  between  two  numbers. 

c.  Learning  the  order  of  the  decades.     The  "teens"  come 
before  the  twenties,  the  fifties  follow  the  forties,  etc. 

d.  Naming  the   number   immediately  following   9,    19,   29, 
etc.,  and  preceding  10,  20,  30,  etc. 

e.  Naming  the  number  in  any  special   decade   that  ends 
with  a  1,  2,  3,  etc. 

f.  Naming  which  of  two  numbers  comes  first  in  the  series 
and  which  of  two  numbers  is  the  greater,  or  less. 

g.  Naming  the  first  number  ending  with  a  1,  2,  3,  etc.,  that 
follows  some  assigned  number,  especially  where  the  answer 
introduces  the  next  decade.     Thus,  name  the  next  number 
after  38  that  ends  with  a  4. 

h.  Adding  1  to  and  subtracting  1  from  any  number.  Add- 
ing 10  to  and  subtracting  10  from  any  number.  Adding  any 
number  less  than  10  to  10,  20,  etc.  Subtracting  3  from  13, 
23,  etc.;  7  from  17,  27,  etc.  Adding  5  to  and  subtracting  5 
from  any  number  ending  with  0  or  5. 

The  pupils  have  perhaps  already  learned  a  few  of  the  ad- 
dition and  subtraction  combinations  in  connection  with  ob- 
ject work,  but  have  not  been  drilled  in  them  as  such. 
Column  Addition:  The  teacher  should  read  Chapter  II.  of  the 
text  carefully  for  the  explanation  of  the  method  used  in 
teaching  the  combinations  and  column  addition.  The  meth- 
od is  to  teach  a  group  of  combinations  and  immediately  ap- 
ply them  in  columns  that  have  been  specially  prepared  for 
the  purpose. 

The  method  of  the  text  is  explained  in  the  following  steps: 

Step  A. — Teach  the  combinations       34326 

+  2      +5      +9      +2      +4 

Read  2  and  3,  not  3  and  2,  in  order  to  conform  to  the  plan 
of  adding  columns,  at  first,  from  the  bottom  up.  Place  the 
pointer  on  the  upper  figure  as  the  numbers  are  added.  No- 
tice that  the  lower  figure  of  the  second  combination  is  the 
sum  in  the  first  and  that  the  lower  figure  of  the  third  is  the 
sum  in  the  second.  See  Step  C  below  and  note  that  the 
column  is  built  according  to  this  arrangement.  The  answer 
may  be  under  the  combinations  at  first,  but  for  drill  pur- 
poses the  answers  should  not  be  written.  Teach  the  reverses 
of  these  combinations  and  drill  on  both  sets,  taking  the  com- 
binations in  any  order —  94225 
+  3  +6  +2  +3  +4 

Step  B  (The  most  important  step). — Establish  the  rela- 
tions that  2  and  3  are  5,  12  and  3  are  15,  22  and  3  are  25,  etc. 
That  is,  adding  3  to  any  number  ending  with  a  2  gives  the 
next  number  ending  with  a  5.  These  may  be  written  at  first: 

34326  34326 

+  12      +15      +19      +12      +14;         +22      +25      +29      +22      +24; 


ARITHMETIC  15 

etc.     In  drills,  do  not  write  the  tens'  figures  nor  the  sums. 

34326 

(a)  -|-2     +5     +9     +2     +4        Give  (a)  as  it  stands.     In 

—      —      —      —        (b),   say    13,   22;     16,  20; 

12,  14;   etc.  Return  to  (a) 

94225         and  say  22,  25; 

(b)  Reverses:    +3     +6     +2     +3     +4         25,    29;    29,  32; 

—      —      —      —      —        etc.     In  (b),  33, 
42;    36,    40;    32, 

34;  etc.  Again,  the  40's  with  (a),  the  50's  with  (b),  and  so 
on  up  to  120.  Next  give  (b)  as  it  stands,  use  the  "teens" 
with  (a),  the  20's  with  (b),  and  so  on  up  to  120.  Extend  the 
work  beyond  120.  This  is  the  fundamental  drill  for  Step  B, 
since  it  uses  each  combination  in  every  decade.  A  variation 
of  this  drill  with  (a)  may  be  used  just  before  or  while  the 
columns  are  being  built.  Thus,  in  (a)  say  5;  9;  12;  12,  14; 
14,  20;  22,  25;  25,  29;  29,  32;  32,  34;  34,  40;  42,  45;  45,  49; 
49,  52;  52,  54;  54,  60;  and  so  on  through  the  decades.  Again, 
repeat  this,  omitting  the  repetition  of  the  sums  that  give  the 
next  lowest  number.  Thus,  say  5,  9,  12,  14,  20;  25,  29,  32,  34, 
40;  etc.  Next  erase  the  lower  numbers  of  (a)  with  the  ex- 
ception of  the  first  combination  and  add  as  just  given. 

34326  When    the    column   is   built 

+  2  from    this    arrangement,   by 

writing  the  4,  3,  2,  6,  in  turn 
above  the 
3 
2,        we  have  the  column  described  in  Step  C  that  follows. 


Step  C. — The  columns  are  built  from  the  bottom  up,  using 
the  combinations  in  Step  A.  Columns  are  added  from  the 
bottom  up  until,  in  the  High  Third,  all  the  combinations  are 
learned  and  applied  in  columns.  The 

teacher  places  the  pointer  beside  the  3  and  the  6 

pupil  says  5.     She  then  places  the  pointer  beside  2  2 

the  4  and  the  pupil  says  9.    Do  not  name  the  first  333 

number  in  the  column,  but  give  the  sum  of  the  4444 
first  two  numbers.  The  combinations  in  Step  A  3333 
should  be  on  the  board  for  reference.  2222 

In  adding  a  column,  never  let  a  pupil  pause  to 
think,  for  in  that  case  he  is  probably  counting  to  get  the 
answer.  If,  in  adding  a  column,  a  pupil  has  said  29,  for  ex- 
ample, and  stops,  the  next  number  being  a  3,  ask  him  for 
the  sum  of  9  and  3.  Then  ask  for  19  and  3,  29  and  3,  etc. 
Whenever  a  pupil  stops  in  a  column  and  after  he  has  been 
corrected,  have  him  begin  again  at  the  bottom,  as  many 
times  as  necessary,  to  carry  him  smoothly  past  the  weak 
place.  It  may  be  necessary  to  caution  some  pupil  not  to 
look  at  a  figure  in  the  column  after  it  has  been  added  in. 
This  will  prevent  the  substitution  of  a  wrong  sum  with 
which  to  continue  the  adding.  Thus,  if  the  pupil  has  come 
to  a  3  in  the  column  and  has  said  25,  the  sum  up  to  that 


16  ARITHMETIC 

point,  there  is  danger  that  he  may  add  the  next  figure  in 
the  column  to  23  instead  of  25  on  account  of  looking  at  the 
3  too  long.  Another  corrective  for  helping  some  pupil  in 
adding  in  the  next  figure  is  to  place  beside  the  column  the 
sum  up  to  that  point.  This  device  should  be  used  sparingly, 
since  pupils  should  not  be  encouraged  to  employ  a  crutch 
that  hinders  speed  and  spoils  the  form  of  the  work. 

A  good  drill  is  secured  by  the  teacher's  naming  a  number, 
say  32,  and  writing  a  number,  3,  for  example,  on  the  board. 
The  pupil  answers  35.     The  drill  should  be  rapid  to  be  ef- 
fective.    This  drill  gives  the  pupils  practically  the  same  ex- 
perience which  they  encounter  in  adding  a  column,  the  hear- 
ing or  thinking  of  one  number  and  the  seeing  of  a  second 
number  to  be  added  to  the  first.     A  good  drill  in  Step  B  is 
for  the  teacher  to  write,  for  example, 
on  the  board.     The  answer  is  given.     Erase  the  2 
+  2  and  substitue  in   turn   12,  22,   32,  etc.     Next  vary 

from  the  serial  order,  using  for  the  lower  number 
92,  42,  62,  etc.     In  some  cases  it  may  aid  to  have 
the  children  count  serially,  2  and  3  are  5,  12  and  3  are  15, 
22  and  3  are  25,  32  and  3  are  35,  etc. 

Do  not  teach  Step  C,  the  addition  of  the  column,  until  the 
pupils  have  thoroughly  memorized  the  combinations  in  Step 
A  and  respond  readily  in  the  drills  in  Step  B.  The  pupils 
should  be  taught  to  study  Step  B  alone.  Then  let  them 
write  the  proper  decades  in  the  lower  numbers  and  write 
the  sums  for  the  inspection  of  the  teacher.  The  pupils 
should  understand  perfectly  the  relation  between  the  col- 
umn in  Step  C  and  the  arrangement  of  the  combinations 
in  Step  A.  They  may  also  be  taught  to  make  their  own 
columns  from  Step  A  and  hence  secure  a  very  effective 
means  of  studying  alone. 

Do  not  have  at  first  more  than  six  numbers  in  the  single 
or  double  columns.  The  higher  numbers  can  be  reached  by 
adding  some  multiple  of  10  to  the  lowest  figure  in  the  single 
column,  as  in  Step  B.  Thus,  if  8  is  the  lowest  figure,  think 
58,  78,  or  128  in  its  place  and  add. 

Supplement  from  the  Handbook  for  exercises  in  addition 
and  subtraction.  After  a  good  start  is  made  in  Lesson  A  in 
addition  have  the  pupils  memorize  and  apply  the  corresponding 
subtraction  combinations. 

To  the  addition  combination  4  99 

+  5  correspond  — 4  and  — 5 

In  subtracting,  follow  the  plan  of 

the  text  in  using  the  Austrian,  or  addition,  method,  where  one 
asks,  4  and  how  many  are  9?  Or  we  may  say,  "4  to  make  9, 
five."  This  form  is  good  on  account  of  its  retaining  the  mak- 
ing up  idea  and  at  the  same  time  giving  some  distinction  be- 
tween verbal  addition  and  subtraction.  Either  of  these  forms 
is  to  be  used  in  the  written  work  that  is  to  follow.  In  oral 
work  use  these  and  also  the  following:  12  less  3  is  what?  What 
is  the  difference  between  12  and  3?  Make  up  story  problems 
to  correspond.  .Objects  may  prove  of  service  here.  Do  not 
neglect  to  subtract  like  numbers  in  the  written  work,  using  the 


ARITHMETIC  17 

Austrian  method.     Provide  for  the  addition  of  1  and  0  In  col- 
umn addition. 

After  the  addition  of  single  columns  is  well  in  hand  and 
also  the  subtraction  of  numbers  of  two  figures  each,  begin  the 
addition  of  two  and  three  columns,  which  almost  immediately 
involves  the  idea  of  "carrying." 

Use  the  following  language  form  in  column  addition:     The 

pupil  says,  in  adding  the  first  column:  12,  14.  Write 
42  down  the  4  and  add  the  1  to  the  next  lowest  figure. 

23  3,  5,  9,    Write  down  the  9.     The  sum  of  the  numbers  is 

29  94.      This   explanation    may   be   discontinued   after   all 

the   class   understand    the   process    of   "carrying."     In 
4  adding  the  first  column,  the  sum  must  be  visualized. 

How  many  figures  in  14?  If  we  had  but  one  column 
we  should  write  down  both  figures  in  the  sum  for  that  column, 
but  since  there  is  a  second  column  the  1  of  the  14  is  added  to 
the  second  column.  The  1  ten  of  the  14  belongs  in  the  ten's 
column. 

Defer  the  hard  case  in  subtraction  until  the  Low  Third. 
Also  defer  written  tabulations  by  the  pupils,  such  as  is  given 
on  p.  55  of  the  text.  Care  must  be  used  in  making  up  exam- 
ples in  addition  and  subtraction  not  to  introduce  combinations 
not  already  memorized.  The  work  of  the  High  Second  in  ad- 
dition and  subtraction  is  restricted  to  Lesson  A  of  the  text 
supplemented  by  examples  from  the  Handbook  and  others 
made  up  by  the  teacher. 


LOW  THIRD. 


Text. — California  State  Series,  "First  Book  in  Arithmetic,"  pp. 
55-80.  Supplementary  exercises  from  Handbook,  Part  I. 
Text  in  the  hands  of  the  pupils. 

New  Phases  of  Formal  Processes. — The  hard  case  in  subtrac- 
tion, deferred  from  Lesson  A  in  the  text.  The  addition  and 
subtraction  combinations  in  Lessons  B,  C,  and  D  of  the  text 
and  applications  in  column  addition  and  subtraction. 

Language  Forms. — In  the  hard  case  in  subtraction.  Written 
tabulations  in  addition  and  subtraction  problems. 


While  the  formal  number  processes  are  being  developed, 
keep  alive,  by  reviews  throughout  the  term,  the  work  in  com- 
parative magnitudes  and  the  related  objectice  work  in  frac- 
tions. Review  telling  time,  the  reading  of  the  Roman  numerals 
up  to  XIL,  and  the  relations  already  learned  in  liquid  measure 
and  linear  measure.  The  text  provides  for  most  of  this  work. 
Continue  reading  and  writing  numbers,  using  numbers  as  high 
as  two  periods.  Insist  on  straight  columns  and  good  figures. 
There  should  be  occasional  speed  drills  in  writing  numbers. 
Test  the  class  in  enumerating  the  orders.  Introduce  the  writing 
of  dollars  and  cents,  using  the  decimal  point.  The  main  line 
of  emphasis  for  this  grade  should  be  on  addition  and  subtrac- 


18  ARITHMETIC 

tion.  Before  taking  up  Lesson  B  of  the  text,  review  the  addi- 
tion and  subtraction  in  Lesson  A.  Do  not  leave  this  until  col- 
umns are  added  with  great  accuracy  and  smoothness  and  with 
considerable  speed.  Follow  the  same  plan  of  teaching  Lessons 

B,  C,  etc.,  as  was  employed  in  Lesson  A,  using  Steps  A,  B,  and 

C.  See  High  Second.    Follow  the  suggestion  given  in  the  High 
Second  to  utilize  the  higher  numbers  in  a  column  by  calling 
the  lowest  number  of  a  single  column;   for  example,  47    where 
the  figure  actually  written  is  7.     Add  orally  two  numbers  of 
two  figures  each,  as  on  pp.  60  and  66   (not  p.  76)  of  the  text. 
The  method  is  that  of  adding  the  tens  first.     The  examples  for 
this  grade  are  of  the  type  40  and  20,   46  and  20.     It  may  be 
necessary  to  write  these  at  first  in  column  form.     If  so,  insist, 
as  in  the  second  example,  that  the  pupil  says  46,  66.     It  should 
be  the  aim  to  add  such  numbers  mentally  without  forming  a 
mental  image  of  one  number  under  another.  Any  image  formed 
should  relate  to  the  numbers  in  series. 

More  time  may  be  given  to  seat  work  than  in  the  previous 
grade.  Columns  involving  combinations  not  yet  thoroughly 
learned  should  be  studied  at  the  board  under  the  direction  of 
the  teacher.  The  tabulation  of  written  problems  may  now  oc- 
casionally be  required  of  the  pupils.  The  following  tabulation 
of  a  concrete  problem  is  sufficient  for  this  grade: 

28  boys. 
+  26  boys 

54  boys 

Require  the  following  language  form  in  the  hard  case  in 
subtraction  until  the  results  show  that  the  process  will  not  b? 
readily  forgotten: 

92         Since  3  is  greater  than  2,  no  number  added  to  3  can 
— 33        give  2.     Hence  we  think  12  in  the  place  of  the  2.     3 
—        and  9  are  12.    Write  down  the  9  and  add  1  to  the  next 
9         lower  figure,  making  it  4.    4  and  5  are  9.    Write  down 
the  5.     The  difference  between  92  and  33  is  59.     In- 
stead of  saying  "3  and  9  are  21,"  "3  to  make  12    nine,"  may  be 
used. 

Flash  cards  on  which  are  written  the  addition  and  subtrac- 
tion combinations  that  have  been  taught  up  to  this  point  will 
prove  of  great  help  in  fixing  the  combinations.  The  number 
charts,  previously  mentioned,  should  be  on  the  wall  for  ready 
reference  in  case  any  pupil  has  forgotten  the  position  of  any 
number  in  the  series.  Impress  on  the  minds  of  the  children 
some  conception  of  the  magnitude  of  the  numbers  they  are 


ARITHMETIC  19 

using.     Are  there  500  children  in  this  room?     Are  there  100? 
Are  there  20? 

In       making       up       examples       in       subtraction       where 
"carrying"       is       involved,       the      teacher      must      use      care 
to       bring       in       no       new       combinations.       Thus       in         52 
the  second  column  depends  primarily  upon  the  addition    — 23 
combination   3   and  2   are   5.     In  using  the   "borrowing" 
method,  not  employed  in  this  course,  the  second  column 
would  depend  upon  the  fact  that  4  less  2  is  2. 

At  the  conclusion  of  Lesson  B  of  the  text,  which  is  the  limit 
of  the  work  for  the  Low  Third,  there  have  been  taught  20  addi- 
tion combinations,  their  reverses,  and  the  corresponding  com- 
binations in  subtraction. 


HIGH  THIRD. 


Text. — California  State  Series,  "First  Book  in  Arithmetic,"  pp. 
81-114.  Supplementary  exercises  from  the  Handbook,  Part  I. 

New  Phases  of  Formal  Processes. — The  addition  and  subtrac- 
tion combinations  given  in  Lessons  E,  F,  G,  and  H  of  the 
text.  Column  addition  and  subtraction  to  correspond.  This 
completes  the  learning  of  the  combinations.  Multiplication 
and  division  tables  and  related  examples  given  in  Lessons 
A  and  B  on  multiplication  and  division  in  the  text.  Multi- 
plier of  one  figure,  multiplicand  as  many  as  three  figures. 
Short  division,  without  remainder  in  steps  or  at  the  end. 

Language  Forms  and  1  echnical  Terms. — Language  form  in 
multiplication  where  "carrying"  is  involved.  Tabulations  in 
addition  and  subtraction  problems.  Use  of  terms  product, 
quotient,  area.  Use  of  symbols  +,  — ,  =,  in  the  equation. 


Review  column  addition  and  subtraction,  employing  the 
combinations  learned  in  previous  grades.  Since  the  addition 
and  subtraction  combinations  are  completed  in  this  grade,  the 
work  in  column  addition  and  subtraction  should  be  thoroughly 
mastered.  Whatever  the  rate  of  speed  of  the  individual  pupil 
in  adding  columns,  he  should  add  with  a  certain  rhythm  that 
only  comes  with  a  thorough  knowledge  of  the  combinations  and 
the  place  of  the  numbers  in  the  series.  Towards  the  end  of 
the  semester,  after  all  the  combinations  have  been  used  in 
columns,  have  columns  added  from  the  top  down.  Then  prove 
results  by  adding  from  the  bottom  up.  Flash  cards  on  which 
are  written  the  combinations  will  be  found  helpful  in  quick 
drills.  Also  the  circle  device,  common  in  most  texts,  will  serve 
the  same  purpose.  An  excellent  drill  is  secured  by  having  the 
pupils  write  a  list  of  numbers  in  a  column  at  the  left  of  a  ver- 
tical line.  Over  a  line  at  the  top  write  the  number  (single 


20  ARITHMETIC 

figure)  to  be  added  to  the  numbers  in  the  column  or  subtracted 
from  them.  Place  answers  in  a  vertical  column  at  the  right. 
There  is  an  excellent  opportunity  here  for  speed  contests.  An- 
other good  review  of  the  combinations  is  secured  by  asking  for 
all  the  addition  combinations  that  give,  say,  15.  Give  mental 
drills  like:  4,  add  8,  add  4,  subtract  9,  add  3,  subtract  2,  add  30, 
etc.  Play  store,  using  toy  money  and  making  change  acced- 
ing to  business  methods  (See  High  Second).  Review  reading 
and  writing  numbers  of  two  periods.  Enumerate  the  orders — 
units,  tens,  etc.  Point  out,  in  305,  964,  the  figure  in  hundreds' 
place.  In  hundred-thousands'  place.  Continue  the  addition 
of  two  numbers  of  two  figures  each  after  the  manner  suggested 
in  the  Low  Third.  Precede  those  given  on  p.  76  of  the  text  by 
numbers  used  in  making  purchases.  Thus,  add  45  and  15,  45 
and  35.  In  the  latter,  the  pupil  says  45,  75,  80  or  40,  70,  80,  or 
perhaps  he  may  learn  to  say  50,  80.  Pupils  should  be  able  to 
add  quickly,  mentally,  numbers  like  13  and  17,  14  and  16,  24 
and  16. 

The  class  should  occasionally  be  drilled  in  tabulating  addi- 
tion and  subtraction  problems  according  to  the  form  given  in 
the  Low  Third  and  also  on  p.  55  of  the  text. 

Before  beginning  the  work  in  multiplication  and  division, 
review  the  measure  contents  of  the  early  numbers  objectively. 
See  work  for  the  High  First  and  the  Low  Second.  Read  Chap- 
ter II.  of  the  text,  which  gives  the  authors'  plan  of  developing 
multiplication  and  division.  Follow  the  plan  of  the  text  with 
respect  to  learning  and  applying  the  multiplication  and  division 
tables.  Notice  that  parts  of  the  tables  are  learned  and  then 
applied.  Thus  in  Lesson  A  on  multiplication  the  pupil  learns 
the  products:  two  2's,  three  2's,  four  2's,  three  3's,  and  the  re- 
verses. After  these  are  applied  in  examples,  the  correspond- 
ing division  tables  are  taught  and  applied.  The  same  plan  holds 
In  Lesson  B  in  multiplication,  where  the  pupil  learns  the  prod- 
ucts: five  2's,  six  2's,  four  3's,  five  3's,  and  the  reverses. 

The  pupils  should  identify,  for  example,  five  3's  are  15,  with 
the  result  of  adding  3  five  times  or  counting  by  3's  as  far  as  15 
This  relation  between  addition  or  serial  counting  and  multi- 
plication should  be  brought  out  from  time  to  time  while  the 
tables  are  being  developed,  not  so  much  as  an  aid  for  memor- 
izing the  tables,  for  this  may  not  generally  assist,  but  as  a 
means  of  associating  equivalent  mathematical  facts  with  re- 
spect to  the  number  series.  Notice  that  when  saying  the  mul- 
tiplication table  of  3's  one  says  five  3's  are  15  and  when  saying 
the  table  of  5's,  three  5's  are  15.  Notice  that  it  is 
not  the  plan  of  the  text  to  have  the  tables  "said,"  since 


ARITHMETIC  21 

some  of  the  reverses  bring  in  isolated  products  from  the  higher 
tables.  Thus  in  Lesson  B  on  multiplication  the  pupil  learns 
the  3's  as  far  as  five  3's.  The  reverse  of  five  3's  is  three  5's,  the 
first  of  the  5's  that  has  been  learned. 

Use  the  following  language  form   in  multiplication,   where 

326 

"carrying"  is  involved,  until  the  process  is  established:          X2 

Two  6's  are  12.    Write  down  the  2  and  carry  the  1.    Two  2 

2's  are  4  and  1  are  5.     Etc.     652  is  the  product  obtained 
by  multiplying  326  by  2.     Do  not  permit  the   writing  of  the 
numbers  "carried." 

The  work  in  fratcions  in  the  High  Third  is  a  continuation 
of  the  work  of  the  previous  grades.  Halves,  fourths,  eighths, 
thirds,  and  sixth  are  studied  in  connection  with  diagrams  and 
other  objects.  Memorize  the  relations  suggested  in  the  text, 
require  the  readnig  and  writing  of  the  fractions  studied,  but 
do  not  give  examples  in  the  addition  and  subtraction  of  frac- 
tions, except  to  fill  in  the  proper  values  when  relations  are  ex- 
pressed in  the  equation  form,  thus:  1/2  -f  1/4  =  x;  3/4  +  x  =  1. 
What  is  the  fraction  that  should  be  in  the  place  of  x? 


LOW  FOURTH. 


Text. — California  State  Series,  "First  Book  in  Arithmetic,"  pp. 
115-149.  Supplementary  exercises  from  the  Handbook, 
Part  I. 

New  Phases  of  Formal  Processes. — Lessons  C,  D,  E,  F,  and  G 
of  the  text  in  multiplication  and  short  division,  applying  the 
parts  of  the  tables  learned.  Multiplication  by  numbers  of 
two  figures.  In  short  division,  remainders  in  the  steps  and 
at  the  end.  Use  of  fractional  parts  in  problems.  The  indi- 
cated fraction  as  an  aspect  of  division. 

Language  Forms  and  Technical  Terms. — Language  forms  in 
the  mechanics  of  multiplication  where  the  multiplier  has 
more  than  one  figure.  In  short  division,  where  there  are  re- 
mainders, both  in  the  steps  and  at  the  end.  Oral  analysis 
of  multiplication,  division,  and  partition  problems,  includ 
ing  problems  of  one  and  two  steps.  Written  tabulations  for 
the  same.  Use  of  terms  ratio,  multiplier,  multiplicand,  divi- 
dend, divisor. 


Review  column  addition  throughout  the  grade  with  the  idea 
of  gaining  greater  rapidity,  but  not  at  the  expense  of  accuracy. 
Add  columns  in  both  directions.  Encourage  pupils  to  group 
when  adding,  especially  the  numbers  that  give  10.  Follow 
carefully  the  suggestions  of  the  text  in  this  respect.  Drill  in 


22  ARITHMETIC 

adding  two  numbers  of  two  figures  each  mentally  as  suggested 
in  the  text.  See  suggestions  for  the  High  Third.  Also  drill  in 
subtracting  by  the  addition  method  after  the  manner  of  mak- 
ing change.  Ex.:  How  many  years  have  elapsed  from  1888  to 
1911?  Solution:  2,  12,  23.  The  last  number  is  the  answer.  In 
subtracting  39  from  76  we  say:  1,  37.  The  1  is  the  number  that 
must  be  added  to  39  to  give  40.  The  37  is  the  result  of  adding 
the  1  to  36,  which  is  the  number  that  must  be  added  to  40  to 
give  76 

Continue  the  learning  of  the  multiplication  and  division 
tables  and  apply  in  examples.  Use  short  division  only.  Follow 
the  plan  of  the  text.  By  the  end  of  the  Low  Fourth,  25  prod- 
ucts, their  reverses,  and  the  corresponding  division  facts  will 
have  been  memorized  and  used  in  examples.  Flash  cards  and 
the  circle  device  are  excellent  aids  in  fixing  the  tables.  Insist 
on  neat  and  orderly  work  in  all  examples,  especially  in  multi- 
plication where  the  multiplier  has  more  than  one  figure. 

Use  language  forms  in  the  mechanics  of  multiplying  by 
numbers  of  two  figures  and  in  short  division  where  there  are 
remainders  in  the  steps  and  at  the  end.  The  text  gives  models. 
It  is  advisable  to  conclude  the  language  form  in  multiplication 
by  saying,  for  example,  after  multiplying  457  by  20,  that  9140 
is  the  product  obtained  by  multiplying  457  by  20.  This  em- 
phasizes in  the  mind  of  the  pupil  what  he  has  accomplished. 
Follow  a  similar  plan  in  interpreting  the  quotient  in  exam- 
ples in  division.  Have  analyses  given  in  problems  of  one  step 
and  then  of  two  steps.  The  text  gives  models.  The  teacher 
must  use  her  discretion  in  the  continuance  of  language  forms 
in  mechanical  work  and  of  analyses  of  problems.  A  main  ob- 
ject of  the  description  or  explanation  is  to  create  correct  think- 
ing. Use  objective  illustrations  to  help  make  clear  the  solu- 
tion of  problems,  especially  where  the  fractional  idea  is  in- 
volved. In  all  concrete  problems,  be  ready  to  bring  up  in  the 
minds  of  the  pupils  images  of  the  numbered  materials  men- 
tioned in  the  problems,  whether  quarts,  yards,  or  pairs  of  shoes. 
Defer  until  the  High  Fifth  inverse  problems  of  the  type — "If 
2/3  of  a  yard  of  cloth  costs  18c,  what  is  the  cost  of  a  whole 
yard?"  Hence  omit  p.  150. 

The  idea  of  the  fraction  should  be  associated  with  problems 
in  partition.  Thus,  what  is  the  cost  of  1/3  of  a  yard  of  cloth 
if  a  whole  yard  costs  15c?  Or,  what  is  the  cost  of  1  chair  if  4 
chairs  cost  $12?  Emphasize,  as  does  the  text,  that  2)12,  12/2, 
and  12-i-2  all  indicate  the  same  operation.  Each  means  either 
12  divided  by  2  or  1/2  of  12,  according  to  the  nature  of  the 
problem.  Advance  in  the  study  of  fractions  is  made  by  ex- 


ARITHMETIC  23 

pressing  division  examples  in  the  form  of  8/2,  11/2,  and  re- 
ducing to  whole  or  mixed  numbers.  These  are  to  be  worked 
first  as  examples  in  division.  Pupils  should  be  able  to  tell 
what  part  of  18ft.  are  3ft.  Use  the  term  ratio.  What  is  the 
ratio  of  3ft.  to  18ft.?  18ft.  to  3ft.?  There  should  be  much 
oral  work  in  the  simple  fractional  relations. 


HIGH  FOURTH. 

Text. — California  State  Series,  "First  Book  in  Arithmetic,"  pp. 
151-187.  Supplementary  exercises  from  the  Handbook, 
Part  I. 

New  Phases  of  Formal  Processes. — Multiplication  and  division 
tables  completed  and  applied.  Long  division  introduced  and 
completed. 

Language  Forms. — Analyes  of  remaining  types  of  problems  in- 
volving multiplication,  partition,  and  division.  Language 
form  in  long  division. 


Continue  the  reviews  in  column  addition  and  subtraction 
along  the  lines  suggested  in  the  Low  Fourth.  Review  the  parts 
of  the  multiplication  and  division  tables  learned  up  to  this 
point  and  apply  in  examples. 

Complete  the  multiplication  and  division  tables,  as  outlined 
in  Lessons  H-M  in  the  text  and  apply  in  examples  and  con- 
crete problems.  Conclude  the  application  of  the  multiplication 
tables  by  using  11  and  12  as  multipliers  and  based  upon  the 
tables  of  ll's  and  12's.  Use  Case  I.  of  the  text's  special  method 
for  determining  the  quotient  figure  in  long  division.  Also  use 
the  first  principle  under  Case  II.  It  may  not  be  advisable  to 
use  the  other  phases  of  Case  II.  The  text  gives  a  model  for  a 
language  form  to  be  used  in  long  division.  Great  care  should 
be  exercised  in  neatness  and  form  in  long  division.  The  quo  • 
tient  figures  should  be  exactly  over  the  proper  figures  in  the 
dividend.  The  figures  in  the  dividend  when  "brought  down" 
to  the  remainders  should  be  directly  under  their  positions  in 
the  dividend.  This  makes  it  easy  to  tell  which  figure  should 
next  be  "brought  down"  and  avoids  the  practice  of  checking 
off  the  dividend  figures  as  fast  as  they  are  used  in  the  re- 
mainders. Do  not  permit  the  writing  in  of  little  figures  in  the 
dividend  in  short  division. 

Have  constant  reviews  of  the  multiplication  and  division 
tables.  Use  flash  card  sand  the  circle  device.  Begin  the  reci- 
tation period  with  the  drill — 4  times  5,  add  4,  divide  by  6,  multi- 
ply by  9,  subtract  7.  Use  the  drill  suggested  in  the  High  Third 
for  memorizing  the  addition  and  subtraction  combinations, 


24  ARITHMETIC 

writing  a  series  of  numbers  in  a  column  and  placing  above  the 
column  the  number  which  is  to  be  used  either  as  the  multiplier 
or  divisor.  Review  the  combinations  in  the  same  way.  Where 
a  record  of  mental  drill  is  desired,  have  the  pupils  write  their 
answers  on  paper.  Test  the  pupils  on  multiplication  and  di- 
vision by  asking  for  two  numbers  whose  product  is,  say,  24. 
Name  all  the  divisors  of  24.  This  drill  extended  through  the 
series  1-100  will  be  found  productive  of  good  results. 

Continue  the  analysis  of  problems  according  to  the  plans 
of  the  text.  Do  not  neglect  to  refer  to  objects  and  real  experi- 
ences within  the  knowledge  of  the  pupils.  Children  cannot  de- 
scribe things  that  they  do  not  see  or  image.  See  suggestions 
in  the  Low  Fourth.  Defer  until  the  High  Fifth  inverse  prob- 
lems of  the  type  referred  to  in  the  Low  Fourth. 

Continue  the  concrete  work  in  fractions  as  in  the  Low 
Fourth.  The  pupils  should  understand  the  meaning  of  frac 
tions  as  brought  out  in  the  analysis  of  problems  involving  par- 
tition. By  the  end  of  the  High  Fourth  the  class  should  under- 
stand examples  like:  How  many  fourths  of  a  circle  in  1  3/4 
circles?  1/2  of  a  pie  is  how  many  fourths  of  a  pie?  4  2/5  is 
how  many  fifths?  No  rules  are  yet  to  be  taught.  The  pupils 
find  the  number  of  fifths  in  4  2/5  by  first  learning  that  in  a 
whole  thing  there  are  five  5ths.  In  4  wholes  there  are  then 
twenty  5th.  Also  continue  the  reduction  of  improper  fractions 
to  whole  or  mixed  numbers,  working  the  exercises  at  first  as 
examples  in  division. 

Sum  up  with  the  pupils  that  which  they  have  learned  about 
denominate  numbers  in  the  earlier  grades.  Make  the  work 
concrete,  using  the  actual  liquid  measures  and  the  measures 
of  length  in  connection  with  simple  examples.  Teach  and  ap- 
ply the  completed  table  of  linear  measure,  given  on  p.  174  of 
the  text.  Also  develop  the  table  of  square  measure.  Make 
drawings  to  scale,  such  as  is  given  on  p.  176  of  the  text.  Meas- 
ure the  tennis  court  and  make  a  drawing  to  scale.  Measure 
some  portion  or  all  of  the  school  garden  and  draw  to  scale. 

By  the  end  of  the  High  Fourth,  the  pupils  should  be  thor- 
oughly grounded  in  the  mechanics  of  addition,  subtraction, 
multiplication,  and  division.  They  should  read  and  write  num- 
bers readily.  They  should  be  able  to  work  and  to  anaiyze  prob- 
lems in  one  and  in  two  steps  involving  processes.  They  should 
know  the  simple  fractional  relations  involving  halves,  fourths, 
eighths,  thirds,  and  sixths.  They  should  be  able  to  reduce  im- 
proper fractions  to  whole  or  mixed  numbers  and  simple  mixed 
numbers  to  improper  fractions.  They  have  memorized  some 
of  the  simple  addition  and  subtraction  facts  in  fractions.  They 


ARITHMETIC  25 

can  read  and  write  the  fractions  they  have  studied.  They  know 
the  relations  between  the  pint  .quart,  and  gallon.  They  know 
the  tables  of  linear  and  square  measure  and  can  draw  simple 
draiwngs  to  scale. 


LOW  FIFTH. 

Text. — California  State  Series,  "First  Book  in  Arithmetic,"  pp. 
188-211;  227-230.  Supplementary  exercises  from  Handbook, 
Part  II.,  of  Bulletin  No.  11,  State  Normal  School,  San  Fran- 
cisco, Cal.,  edited  by  D.  R.  Jones;  and  from  the  advanced 
text.  See  Low  Sixth. 

New  Phases  of  Formal  Processes. — Addition,  subtraction,  and 
multiplication  of  simple  common  fractions  with  necessary 
work  in  reduction — to  lower  terms,  to  higher  terms,  im- 
proper fractions  to  whole  or  mixed  numbers,  and  mixed 
numbers  to  improper  fractions.  Reading  and  writing  of 
decimals.  Addition  and  subtraction  of  decimals. 


Continue  the  review  of  column  addition  and  give  mental 
drills  in  all  the  fundamental  processes.  Add  and  subtract  men- 
tally any  two  numbers  of  two  figures  each  by  methods  sug- 
gested in  grades  three  and  four.  Review  the  multiplication 
and  division  tables  and  apply  in  examples.  Distribute  through- 
out the  term  problems  of  the  character  of  those  given  in  the 
fourth  grade.  Keep  alive  and  extend  the  facility  gained  in 
analysis.  Review  especially  the  analysis  involving  partition 
since  this  work  relates  to  the  study  of  fractions. 

With  respect  to  new  work,  take  up  the  addition,  subtraction, 
and  multiplication  of  common  fractions  in  the  order  named 
employing  familiar  fractions,  such  as  given  in  the  text.  Then 
teach  the  reading  and  writing  of  decimals  as  far  as  thous- 
andths, followed  by  the  addition  and  subtraction  of  decimals. 
The  division  of  common  fractions  will  precede  the  multipli- 
cation and  division  of  decimals  in  the  High  Fifth.  This  ar- 
rangement secures  for  the  pupil  an  early  understanding  of 
decimals  and  enables  him  to  take  up  division  of  common  frac- 
tions with  less  confusion  than  if  it  followed  immediately  the 
multiplication  of  common  fractions.  Teach  all  the  cases  in  the 
multiplication  of  common  fractions,  including  pp.  216  and  217 
of  the  text.  Make  early  use  of  concellation  In  connection  with 
the  multiplication  of  common  fractions. 

The  work  of  the  previous  grades,  which  has  been  oral  and 
based  on  the  use  of  objects,  should  have  secured  for  the  pupils 
a  thorough  understanding  of  the  meaning  of  fractions.  The 
symbolism  of  common  fractions  is  the  main  cause  of  difficulty 
in  this  study.  Teachers  should  spare  no  pains  to  have  pupils 


26  ARITHMETIC 

understand  thoroughly  the  fundamentals  of  the  beginning  work 
in  fractions  by  referring  to  diagrams  and  objects.  The  device 
of  writing,  say,  3/5  in  the  form  3  fifths  secures  a  close  correla- 
tion with  the  spoken  form  and  may  assist  in  making  clear  the 
fundamental  processes.  Make  sure  that  the  pupils  can  read 
fractions  on  the  ruler  and  that  they  can  draw  to  scale.  Such 
work  has  an  application  in  manual  training.  Do  not  teach  here 
a  "method"  of  finding  G.  C.  D.,  the  examples  in  reduction  to 
lower  terms  being  sufficiently  simple  to  reduce  by  successive 
steps  or  by  finding  the  G.  C.  D.  by  inspection.  Neither  should 
a  "method"  of  finding  L.  C.  M.  be  necessary  for  the  addition 
and  subtraction  of  the  common  fractions  used  in  this  grade. 
See  the  Handbook  with  reference  to  this  matter.  Teach  the 
reduction  of  fractions  to  lowest  terms  before  adding  fractions 
whose  answers  need  to  be  reduced.  An  answer  should  not  be 
accepted  as  correct  unless  it  has  been  reduced  to  its  lowest 
terms. 

Precede  the  study  of  decimals  by  a  review  of  the  notation 
and  numeration  of  integers,  emphasizing  the  naming  of  the 
orders  and  periods  and  the  relations  between  the  units  of  the 
various  orders,  that  is,  the  10  times,  1/10  of,  100  times,  1/100 
of,  etc.  Extend  the  principles  here  brought  out  into  decimals. 
The  work  may  be  based  on  United  States  money  or  measure- 
ments with  a  decimal  rule,  tape,  or  chain.  After  the  work  is 
well  in  hand,  relate  the  writing  of  decimals  to  that  of  common 
fractions.  The  practice  of  beginning  decimals  by  saying,  for 
example,  that  7/10  is  also  written  .7  is  open  to  objections,  the 
main  one  being  that  a  misconception  of  the  use  of  the  point 
arises.  It  will  be  sufficient  for  this  grade  to  read  and  wHta 
decimals  having  but  three  places  to  the  right  of  the  point. 
Secure  a  review  of  column  addition  in  the  addition  of  decimals. 

The  problems  in  ratio  and  proportion  (not  formally  treated) 
on  pp.  224  and  225  of  the  text,  should  receive  special  emphasis. 
Any  possible  over  emphasis  of  the  method  of  analysis  may  be 
avoided  by  the  use  of  ratio  and  proportion.  Thus,  why  find 
the  cost  of  1  apple  in  the  problem — If  4  apples  are  worth  5c,  12 
apples  are  worth  c. 


HIGH    FIFTH. 


Text.— California  State  Series,  "First  Book  in  Arithmetic,"  pp. 
212-226;  231-256.  Supplementary  exercises  from  the  Hand- 
book, Part  II.;  also  from  the  advanced  text.  See  Low  Sixth. 

New  Phases  of  Formal  Processes. — Division  of  simple  common 
fractions.  Changing  simple  common  fractions  to  decimal 
hundredths,  and  conversely.  Multiplication  of  decimals. 


ARITHMETIC  27 

Changing  simple  common  fractions  to  per  cents,  and  con- 
versely.    Division  of  decimals. 

Review  especially  the  work  covered  in  common  fractions  in 
the  Low  Fifth.  Review  the  operations  with  whole  numbers 
as  the  needs  of  the  class  require.  Continue  tfle  quick  mental 
drills,  including  the  simpler  operations  with  fractions.  Review 
the  types  of  problems  studied  in  the  Fourth  Grade.  After  tak- 
ing up  the  division  of  common  fractions,  teach  the  type  of  in- 
verse problems  involving  fractional  relations  deferred  from  the 
Low  Fourth.  See  p.  150  of  the  text. 

Interpret  the  principles  involved  in  the  division  of  common 
fractions  by  diagrams,  because  much  of  the  application  work 
in  fractions  has  to  do  with  some  sort  of  objects.  Follow  the 
suggestion  of  the  text  in  dividing  a  mixed  number  by  a  whole 
number  as  in  ordinary  division.  This  plan  is  usually  better 
than  that  of  reducing  to  improper  fractions.  See  that  the  class 
has  facility  in  this  work,  for  in  manual  training  they  are  re- 
quired to  find,  say,  1/4  of  8  1/2.  It  is  much  easier  to  divide 
8  by  4  and  then  1/2  by  4  than  to  reduce  8  1/2  to  an  improper 
fraction  and  then  divide.  Examples  equivalent  to  this  are  re- 
quired later  in  the  sewing  classes,  where  it  is  necessary  "to 
measure  1/2  the  width  of  the  back  plus  1  1/2  in.";  or  "out  from 
1  measure  3  in.  less  than  1/2  (or  1/4)  the  waist  measure." 
Drawing  rectangular  areas  to  scale  secures  a  good  correlation 
between  practice  and  principles. 

Perfect  the  work  in  common  fractions  by  requiring  written 
work  to  be  as  brief  as  possible.  Restrict  the  work  to  simple 
fractions,  such  as  given  in  the  text.  While  percentage  is  not 
taken  up  as  a  topic  for  study  in  this  grade,  follow  the  plan  of 
the  text  in  the  use  of  the  symbol  for  per  cent,  applying  it  in 
such  simple  examples  in  interest  that  can  be  readily  worked 
by  the  class.  Terms  like  "1/2  off"  and  25%  off"  are  seen  daily 
by  the  pupils  in  the  store  windows.  Simple  problems  involving 
these  terms  can  be  readily  worked  in  class.  Follow  the  method 
of  the  text  in  pointing  off  in  the  division  of  decimals.  Require 
the  pupils  to  give  at  first  all  the  steps,  using  a  language  form 
after  the  manner  of  the  work  of  the  lower  grades.  In  this 
grade  restrict  all  work  with  decimals  to  thousandths,  possible 
higher  orders  appearing  only  in  products. 

Conclude  the  work  of  the  High  Fifth  by  selecting  from 
Chapter  VI.,  on  denominate  numbers,  such  tables  and  exercises 
as  constitute  more  of  a  review  than  the  beginning  of  new 
work.  The  table  of  paper  measure  should  be  omitted.  Also 
omit  barrels  and  hogsheads  in  liquid  measure.  The  reading 


ARITHMETIC 

and  writing  of  the  Roman  numerals  should  now  be  reviewed 
and  extended  into  the  higher  numbers.  The  teacher  will  use 
her  judgment  concerning  the  other  topics  of  Chapter  VI.  Much 
depends  upon  the  progress  of  the  class  in  fractions. 


LOW  SIXTH. 


Text. — California     State     Series,     "Advanced    Arithmetic,"    pp. 
1-112.  Supplementary  exercises  from  the  Handbook,  Part  II. 


The  work  of  the  Low  Sixth  is  a  review  and  extension  of  the 
operations  with  whole  numbers  and  fractions,  both  common 
and  decimal.  Review  first  addition,  subtraction,  bultiplication, 
and  division  of  whole  numbers.  If  pupils  are  weak  in  addition, 
drill  on  the  combinations.  If  weak  in  multiplication  and  divi- 
sion, drill  on  the  tables.  The  work  in  common  fractions  in  the 
Fifth  Grade  was  restricted  to  very  simple  fractions.  While 
the  present  work  should  still  be  restricted  to  simple  rather 
than  complex  or  very  small  fractions,  the  pupils  are  expected 
to  work  with  denominators  that  require  the  use  of  a  method 
of  finding  L.  C.  M.  The  inspection  method  will,  however: 
handle  the  great  majority  of  examples.  In  the  Fifth  Grade  the 
work  in  decimals  did  not  employ  orders  beyond  thousandths. 
The  pupils  should  now  be  drilled  on  reading  and  writing  deci- 
mals with  orders  up  to  millionths.  The  higher  orders  should 
appear  in  the  applications  only  in  products  and,  possibly,  in 
dividends. 

Take  up  the  work  topically  as  outlined  in  the  text,  but  keep 
alive  in  the  minds  of  the  pupils,  through  examples,  any  prin- 
ciples learned  in  the  Fifth  Grade  that  will  not  be  systematical- 
ly reviewed  until  the  last  part  of  the  term  or  in  the  High  Sixth, 
for  example,  the  multiplication  and  division  of  common  frac- 
tions. In  this  review  give  more  attention  to  rationalizing  the 
processes  than  in  previous  grades,  where  reasons  were  gener- 
ally not  given  or  required.  Lay  special  emphasis  on  the  deci  - 
mal  system  as  illustrated  in  whole  numbers  and  in  decimal 
fractions,  reviewing  the  operations  with  decimals  with  those 
for  whole  numbers,  as  in  the  text.  Emphasize  oral  and  mental 
arithmetic.  Work  for  speed  in  column  addition,  following  the 
suggestions  for  grouping  given  in  the  previous  grades.  Pupils 
should  multiply  readily  by  11  and  12  and  use  these  numbers 
in  short  division.  Encourage  the  memorizing  of  products  like 
13  times  2,  3,  4,  5;  14  times  2,  3,  4,  5;  15  times  2,  3,  4,  5,  6;  16 
times  2,  3,  4,  5,  6;  17  times  2  and  3;  18  times  2;  19  times  2.  Also 
encourage  memorizing  the  corresponding  division  facts.  Apply 


ARITHMETIC  29 

in  examples.  Teach  usable  short  processes,  such  as  given  in 
the  text.  Estimating  answers  before  solving,  as  on  p.  41  of  the 
text,  is  important  and  should  be  freely  used.  Explain  how  to 
write  a  decimal  to  a  specified  degree  of  accuracy.  Thus,  2.38 
is  written  2.40  or  2.4,  correct  to  tenths.  Also  3.572  becomes 
3.57,  correct  to  hundredths.  Make  use  of  this  practice  in  in- 
terpreting answers. 

The  solution  and  analysis  of  problems  should  be  distributed 
throughout  the  term. 


HIGH  SIXTH. 


Text. — California  State  Series,  "Advanced  Arithmetic,"  pp. 
113-165.  Supplementary  exercises  from  the  Handbook, 
Part  II. 

Complete  the  general  review  for  the  Sixth  Grade,  special 
attention  being  given  to  the  multiplication  and  division  of 
common  fractions,  left  over  from  the  topical  review  of  the  Low 
Sixth.  Emphasize  cancellation.  Continue  the  use  of  mental 
approximations  and  the  writing  of  decimal  answers  to  specified 
degrees  of  accuracy.  See  the  Low  Sixth.  Drill  on  the  frac- 
tional relations  that  will  occur  in  percentage.  Thus,  find  the 
part  one  number  is  of  another  as  well  as  how  many  times  one 
number  is  another.  Use  the  aliquot  parts  for  short  cuts  in 
multiplication  and  division.  Give  abundant  drills  in  changing 
common  fractions  to  decimals  and  decimals  to  common  frac- 
tions, the  latter  always  being  reduced  to  lowest  terms  unless 
otherwise  specified.  Change  both  common  and  decimal  frac- 
tions to  hundredths  so  as  to  prepare  for  the  later  work  in  per- 
centage. 

Work  in  denominate  numbers  has  claimed  attention  from 
the  early  grades.  Conclude  the  work  of  the  High  Sixth  by  re- 
viewing all  the  previous  work  and  that  given  in  the  text  as  far 
as  p.  165.  Emphasize  lumber  measure.  See  the  last  chapter 
in  the  primary  text.  The  Handbook  gives  abundant  exercises. 
The  tables  and  examples  in  denominate  numbers  as  they  are 
gradually  developed  in  the  primary  and  advanced  texts  are 
sufficient  for  ordinary  needs.  Pupils  should  know  where  to 
find  in  their  text  (in  the  appendix)  the  list  of  tables,  including 
those  not  of  common  value.  Omit  the  metric  system,  surveyors' 
measures,  troy  weight,  apothecaries'  weight,  and  apothecaries' 
liquid  measure.  Parts  of  some  questionable  tables  may  be 
taught.  Thus,  if  one  knows  that  16  fluid  ounces  equal  1  pint, 
he  is  in  a  position  to  know  the  size  bottle  to  call  for  at  the 
druggist's.  The  selection  of  problems  in  denominate  numbers 


30  ARITHMETIC 

by  the  authors  of  the  primary  and  advanced  texts  is  well  made. 
While  the  texts  have  a  considerable  number  of  exercises  dis- 
tributed through  the  reviews,  it  is  advisable  to  have  some  con- 
secutive drill  in  denominate  number  problems.  Examples  like 
"Multiply  (or  divide)  4  hhd.  5  bbl.  9  gal.  3  qt.  1  pt.  1  gill  by  5," 
have  little  or  no  practical  value  in  life.  They  are  wisely 
omitted  by  the  text.  The  longer  examples  in  the  addition  and 
subtraction  of  denominate  numbers  should  not  be  worked.  The 
finding  of  differences  between  dates  is  of  practical  value,  es- 
pecially in  interest  examples,  but  the  addition  of  dates  should 
not  be  taught.  Instead  of  having  the  class  work  the  traditional 
exercises  in  denominate  numbers,  it  would  be  more  profitable 
to  have  examples  of  more  immediate  interest  to  the  class.  Thus, 
the  girls  might  be  questioned  on  the  following  table  of  dry 
measure  used  later  in  the  cooking  class: 

If  a  recipe  calls  for  a  certain  amount 

3  tsp    =  1  tbsp          of   flour    and     other     ingredients,    how 

16  tbsp  =  1  cup  many  tablespoons  of  flour  are  used  if 

2  c        =  1  pt  the  recipe   is   reduced   1/3?     Both   the 

2  pt      =  1  qt  boys    and    girls    will    be    interested    in 

problems  involving  local  data.  A  chain 

of  problems  relating  to  a  local  hay  business  may  be  proposed 
and  solved.  Get  data  on  the  cost  of  seeding  an  80-acre  tract 
in  alfalfa.  The  cost  of  keeping  the  ground  in  condition.  The 
cost  of  cutting  the  four  or  more  crops.  The  cost  of  baling. 
Figure  the  profit  after  learning  the  market  price  of  alfalfa  hay. 
Teach  bills  and  accounts,  such  as  the  children  see  at  home. 
Have  a  few  bills  made  out  by  the  pupils.  Which  person  whose 
name  appears  on  the  bill  is  owing  money?  To  whom  is  the 
money  to  be  paid?  Who  receipts  the  bill?  How  is  this  done? 
Teach  how  to  keep  a  cash  account,  such  as  is  illustrated  in  the 
last  chapter  of  the  primary  text.  Use  data  within  the  knowl- 
edge of  the  pupils.  The  keeping  of  accounts  in  connection  with 
the  school  garden  could  well  be  illustrated  here. 


LOW  SEVENTH. 


Text. — California  State  Series,  "Advanced  Arithmetic,"  pp. 
166-206.  Supplementary  exercises  from  the  Handbook, 
Part  II. 


The  work  of  the  Seventh  Grade  consists  almost  entirely  of 
percentage  and  its  business  applications.  The  review  of  the 
Sixth  Grade  that  is  probably  necessary  is  the  changing  of 
common  fractions  to ,  decimal  hundredths  and  the  changing  of 
common  fractions  to  common  fractions  with  denominators  100. 


ARITHMETIC  31 

The  Low  Seventh  takes  up  the  fundamental  mechanical  pro- 
cesses in  percentage  (See  Handbook  for  additional  eercises) 
and  problems  in  profit  and  loss,  commission,  insurance,  taxes, 
and  trade  discount,  reserving  problems  involving  time  for  the 
High  Seventh.  Success  in  percentage  depends  upon  recogniz- 
ing familiar  problems  in  common  and  decimal  fractions  when 
expressed  in  the  language  of  percentage.  The  use  of  the  "x" 
or  "c"  or  some  other  letter  is  recommended  in  the  solution  of 
problems,  this  being  especially  advisable  in  inverse  problems. 
This  does  not  preclude,  however,  the  advisability  of  giving 
mental  work  and  the  simpler  written  problems  in  which  the 
three  types  in  percentage  are  explained  without  the  "x."  Thus, 
after  the  class  has  obtained  considerable  understanding  of  the 
classes  of  percentage  problems,  introduce  the  use  of  letters 
in  a  problem  of  this  type:  "A  man  sold  a  lot  at  a  gain  of  16% 
above  the  cost.  What  was  the  cost  if  the  selling  price  was 
$1,890?"  Write  the  equation,  1.16  of  c  =  $1,890.  Then  find  c. 
The  equation  may  well  be  used  in  problems  where  either  the 
cost  or  per  cent  is  to  be  found.  An  understanding  of  the  equa- 
tion solves  once  and  for  all  the  question  of  the  different  cases 
in  percentage. 

It  should  be  remembered  that  practically  all  percentage 
problems  in  every-day  life  utilize  the  simplest  case  of  per- 
centage, that  involving  multiplication.  The  finding  of  the  per 
cent  one  number  is  of  another  also  has  considerable  applica- 
tion. Pupils  should  be  able  to  work  inverse  problems  like  the 
one  illustrated  above,  but  the  following  in  commission  is  not 
practical  and  should  be  omitted:  "Mr.  Jones  sent  his  agent 
$2000  to  invest  in  flour,  first  taking  out  his  commission  at  3% 
for  buying.  What  were  the  proceeds  and  what  was  the  com- 
mission?" Taxes  and  insurance  have  a  practical  value,  but 
customs  and  duties  do  not  come  within  the  experience  of  the 
ordinary  individual  and  hence  should  be  omitted. 


HIGH  SEVENTH. 

Text. — California    State    Series,    "Advanced    Arithmetic,"    pp. 
207-236. 

Review  the  mechanical  work  in  percentage,  such  as  is  re- 
quired in  the  three  cases  in  percentage.  Also  review  through- 
out the  term  the  business  problems  of  the  Low  Seventh. 

Percentage  is  now  applied  in  problems  where  the  time  ele- 
ment enters.  In  simple  interest,  first  work  problems  in  which 
no  "method"  is  required.  Emphasize  any  one  of  the  proportion 
methods  in  finding  simple  interest.  The  text  gives  two,  the 


ARITHMETIC 

"Method  of  Aliquot  Parts"  and  the  "Sixty  Days'  Method."  The 
cancellation  method  should  also  be  thoroughly  understood.  The 
antiquated  "Six  Per  Cent  Method"  has  little  to  recommend  its 
use  and  need  not  be  taught.  Compound  interest  deserves  at- 
tention, but  no  great  emphasis.  Teach  both  the  Mercantile 
and  the  United  States  Rules  in  partial  payments.  Avoid  prob- 
lems having  more  than  three  payments.  In  work  where  prom- 
issory notes  are  involved,  have  the  class  write  the  notes  and 
try  to  have  the  business  transacted  in  the  class.  Bank  discount 
is  rarely  used  by  the  ordinary  citizen.  Choose  problems  in 
which  the  note  is  discounted  on  the  date  of  issue.  Familiarize  I 
the  class  with  the  different  ways  of  sending  money  and  with 
the  business  that  the  ordinary  citizen  has  to  transact  with  a 
bank. 

Teach  the  mensuration  of  the  familiar  plane  figures  and  \ 
solids  as  outlined  in  the  text  under  Forms  and  Measurements. 
Let  the  class  get  data  for  problems  from  their  own  measure- 
ments either  in  the  school-room  or  in  the  play-ground  or  in 
the  school  garden.  Use  the  compasses  in  the  simplest  geo- 
metric constructions,  such  as  bisecting  angles  and  lines  and 
erecting  perpendiculars.  Some  attractive  designs  may  be  made 
with  compasses. 

Enlarge   upon   the  topics  assigned   to   the  appendix   of  the 
text  according  to  the  needs  and  capacity  of  the  class. 


LOW   EIGHTH. 


Text. — California    State    Series,    "Advanced    Arithmetic,"    pp. 
236-255. 


The  review  of  the  Low  Eighth  includes  the  mechanics  of 
percentage,  percentage  problems,  and  miscellaneous  examples 
in  the  work  covered  in  the  High  Seventh.  In  advance,  teach  I 
longitude  and  time  (correlated  with  geography),  powers  and 
roots,  further  work  in  mensuration,  including  the  application 
of  square  root,  and  the  application  of  proportion  to  similar 
figures,  including  the  finding  of  heights  and  distances.  Before 
leaving  the  review  of  percentage,  bring  together  under 
one  grouping  topics  that  relate  to  investments.  Dis- 
cuss with  the  class  the  various  ways  of  investing 
money.  Promissory  notes  will  be  recalled,  together  with  the 
topics  of  simple  and  compound  interest.  Discuss  the 
features  of  endowment  life  insurance  policies  and  building  and  I 
loan  associations.  The  subject  of  school  and  city  improvement 


ARITHMETIC  33 

bonds  should  be  taken  up  and  explained.  Let  the  class  com- 
pute the  interest  on  any  particular  bond  that  has  been  re- 
deemed or  is  now  drawing  interest.  The  subjects  of  bonds 
and  stocks  studied  as  they  have  been  treated  in  our  texts  in 
the  recent  past  have  little  value  for  the  pupil.  The  operations 
of  a  local  stock  company  can  well  be  explained  to  the  class. 
The  amount  of  any  probable  dividend  or  assessment  can  be 
easily  computed  by  the  class  after  the  teacher  has  shown  a 
certificate  of  stock  and  assigned  a  rate  of  dividend  or  assess- 
ment. The  future  citizen  needs  to  know  more  about  local 
bonds  than  local  stock  companies. 

The  work  of  the  class  may  be  strengthened  by  the  addition 
of  material  from  the  appendix  to  the  text.  In  this  connection 
may  be  mentioned  "Public  Lands,"  which  should  be  correlated 
with  civics,  and  "The  Equation." 


HIGH   EIGHTH. 

The  essential  principles  and  topics  of  arithmetic  have  been 
completed  in  the  Low  Eighth.  The  work  of  the  High  Eighth 
consists,  first,  of  a  review  of  the  most  essential  parts  of  arith- 
metic, and,  second,  of  business  practice  and  the  keeping  of 
simple  accounts  such  as  may  be  needed  by  the  average  person. 

Give  first  attention  in  the  review  to  mental  arithmetic  and 
accuracy  and  speed  in  addition.  Drill  on  the  combinations  and 
the  multiplication  and  division  tables,  as  needed,  to  secure  effi- 
ciency in  the  operations.  Give  rapid  drills  in  the  reading  and 
writing  of  numbers.  The  prime  thing  in  this  general  review 
is  to  give  the  class  mechanical  expertness  in  the  operations 
with  whole  numbers  and  fractions,  both  common  and  decimal. 
Review  the  applications  of  arithmetic  through  lists  of  miscel- 
laneous problems.  Employ  usable  short  cuts  in  all  the  review. 
This  aspect  of  the  term's  work  should  embrace  from  eight  to 
ten  weeks  of  intensive  drill. 

Devote  the  remainder  of  the  term  to  business  practice  and 
related  work.  The  work  of  the  previous  grades  should  be  re- 
viewed and  extended.  Some  of  this  review  relates  to  the  writ- 
ing of  business  paper,  such  as  letters,  bills,  receipts,  and  notes, 
and  to  the  common  business  done  with  a  bank.  Among  other 
things,  the  business  associated  with  the  following  should  be 
understood  by  the  pupils:  Bills,  including  merchandise  tags — 
heading,  items,  receipting  of;  notes — heading,  rate,  mortgage, 
and  other  securities,  other  essential  items;  receipts  and  or- 
ders; checks  and  drafts;  telegrams  and  telephone  messages — 


34  ARITHMETIC 

rates;  money  orders;  paper  money — kinds,  meaning  of.  Pupils 
should  practice  writing  the  business  paper  associated  with  any 
of  the  above  and  should  explain  their  essential  features.  To 
this  should  be  added  the  keeping  of  a  cash  account  and  ac- 
counts with  persons  as  the  minimum  requirements  in  book- 
keeping for  the  grades. 

Have  the  children  write  up  personal  cash  accounts,  the  items 
consisting  both  of  actual  transactions  in  their  own  recent  ex- 
perience and  data  furnished  by  teacher  and  pupils.  Also  teach 
the  class  how  to  keep  an  account  with  some  person.  The  state- 
ment contained  in  a  bill  illustrates  the  fundamental  idea,  but 
require,  in  addition,  the  ledger  form,  one-half  page  for  debits 
and  the  other  half  for  credits,  such  as  is  used  in  ordinary  book- 
keeping. 

The  writing  of  most  of  the  business  paper  and  the  keeping 
of  accounts  may  be  worked  out  in  connection  with  activities 
of  the  pupils.  Let  them  buy  and  sell,  loan  and  borrow,  and  the 
like,  using  school  money  and  merchandise  cards.  One  way  to  , 
unify  the  work  is  to  group  the  activities  about  .a  school  bank, 
with  which  the  pupils  do  business.  In  case  the  bank  idea  is 
not  used,  the  teacher  should  illustrate  the  business  of  deposit- 
ing money  with  and  receiving  and  sending  money  through  a 
bank.  In  like  manner  the  use  of  the  money  order  should  bo 
illustrated.  The  new  regulation  concerning  Postal  Savings 
Banks  should  be  explained  and  the  pupils  encouraged  to  make 
deposits  when  the  local  postoffice  establishes  a  postal  bank. 
In  the  meantime,  the  teacher  or  the  school  bank  can  receive 
deposits  from  the  children  and  keep  the  necessary  accounts. 

A  study  of  household  expenses  may  be  associated  with  the 
study  of  bills  and  accounts.  Classify  the  expenses  of  a  house- 
hold of,  say,  five  persons  for  one  month.  Under  Household 
Maintenance  include  rent,  table,  electricity,  gas,  water,  laun- 
dry, and  miscellaneous  needs  of  the  house.  Under  Personal 
Expenses  of  the  members  of  the  family,  taken  individually, 
include  wearing  apparel,  luxuries  not  shared  in  by  the  rest  of 
the  family,  etc.  Under  Luxuries  include  theaters,  concerts 
confectioners'-,  etc.  Under  Unclassified  Expenses  include  any- 
thing not  included  in  the  other  headings,  such  as  fees,  dues, 
church,  car-fare.  The  teacher  and  the  pupils  supply  the  neces- 
sary data  for  these  items. 

An  opportunity  is  offered  in  such  work  for  the  pupils  to 
learn  the  prices  of  common  articles  of  necessity.  The  ques- 
tions of  values  and  economy  are  immediately  concerned  here. 
Pupils  learn  to  read,  gas,  water,  and  electric  light  meters. 

If  the   teacher   desires,   a  broader   course   in  book-keeping 


ARITHMETIC  35 

may  be  profitably  worked  out  so  as  to  provide  for  the  above- 
mentioned  activities  and  give  the  pupils  practice  in  double- 
entry,  not  a  complete  system  with  three  or  more  books  to  con- 
fuse the  learner,  but  a  simple  plan  which  requires  only  a  blot- 
ter and  a  ledger.  (See  the  manual  furnished  by  the  depart- 
ment of  mathematics  of  the  Normal  School.) 


UNIVERSITY  OF   CAITFOB* 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  S1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


LD  21-100m-7,'39(402 


A 


251757 


26m-6,'12 


